contest_id
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1312 | C | C. Adding Powerstime limit per test2 secondsmemory limit per test256 megabytesinputstandard inputoutputstandard outputSuppose you are performing the following algorithm. There is an array v1,v2,…,vnv1,v2,…,vn filled with zeroes at start. The following operation is applied to the array several times — at ii-th step (00-indexed) you can: either choose position pospos (1≤pos≤n1≤pos≤n) and increase vposvpos by kiki; or not choose any position and skip this step. You can choose how the algorithm would behave on each step and when to stop it. The question is: can you make array vv equal to the given array aa (vj=ajvj=aj for each jj) after some step?InputThe first line contains one integer TT (1≤T≤10001≤T≤1000) — the number of test cases. Next 2T2T lines contain test cases — two lines per test case.The first line of each test case contains two integers nn and kk (1≤n≤301≤n≤30, 2≤k≤1002≤k≤100) — the size of arrays vv and aa and value kk used in the algorithm.The second line contains nn integers a1,a2,…,ana1,a2,…,an (0≤ai≤10160≤ai≤1016) — the array you'd like to achieve.OutputFor each test case print YES (case insensitive) if you can achieve the array aa after some step or NO (case insensitive) otherwise.ExampleInputCopy5
4 100
0 0 0 0
1 2
1
3 4
1 4 1
3 2
0 1 3
3 9
0 59049 810
OutputCopyYES
YES
NO
NO
YES
NoteIn the first test case; you can stop the algorithm before the 00-th step, or don't choose any position several times and stop the algorithm.In the second test case, you can add k0k0 to v1v1 and stop the algorithm.In the third test case, you can't make two 11 in the array vv.In the fifth test case, you can skip 9090 and 9191, then add 9292 and 9393 to v3v3, skip 9494 and finally, add 9595 to v2v2. | [
"bitmasks",
"greedy",
"implementation",
"math",
"number theory",
"ternary search"
] | import sys
def tobase(n, k):
ret = []
while n:
ret.append( n % k)
n //= k
return ret
ncase = int(sys.stdin.readline().strip())
while ncase:
ncase -= 1
n, k = map(int, sys.stdin.readline().split())
a = list(map(int, sys.stdin.readline().split()))
b = [ tobase(i, k) for i in a ]
mlen = max([len(i) for i in b ])
res = []
for i in range(mlen):
m = 0
for x in b:
try :
m += x[i]
except IndexError:
pass
res.append(m)
if res and max(res) > 1:
print("NO")
else:
print("YES")
| py |
1287 | B | B. Hypersettime limit per test3 secondsmemory limit per test256 megabytesinputstandard inputoutputstandard outputBees Alice and Alesya gave beekeeper Polina famous card game "Set" as a Christmas present. The deck consists of cards that vary in four features across three options for each kind of feature: number of shapes; shape, shading, and color. In this game, some combinations of three cards are said to make up a set. For every feature — color, number, shape, and shading — the three cards must display that feature as either all the same, or pairwise different. The picture below shows how sets look.Polina came up with a new game called "Hyperset". In her game, there are nn cards with kk features, each feature has three possible values: "S", "E", or "T". The original "Set" game can be viewed as "Hyperset" with k=4k=4.Similarly to the original game, three cards form a set, if all features are the same for all cards or are pairwise different. The goal of the game is to compute the number of ways to choose three cards that form a set.Unfortunately, winter holidays have come to an end, and it's time for Polina to go to school. Help Polina find the number of sets among the cards lying on the table.InputThe first line of each test contains two integers nn and kk (1≤n≤15001≤n≤1500, 1≤k≤301≤k≤30) — number of cards and number of features.Each of the following nn lines contains a card description: a string consisting of kk letters "S", "E", "T". The ii-th character of this string decribes the ii-th feature of that card. All cards are distinct.OutputOutput a single integer — the number of ways to choose three cards that form a set.ExamplesInputCopy3 3
SET
ETS
TSE
OutputCopy1InputCopy3 4
SETE
ETSE
TSES
OutputCopy0InputCopy5 4
SETT
TEST
EEET
ESTE
STES
OutputCopy2NoteIn the third example test; these two triples of cards are sets: "SETT"; "TEST"; "EEET" "TEST"; "ESTE", "STES" | [
"brute force",
"data structures",
"implementation"
] | import sys
from math import sqrt, gcd, factorial, ceil, floor, pi, inf, isqrt, lcm
from collections import deque, Counter, OrderedDict, defaultdict
from heapq import heapify, heappush, heappop
#sys.setrecursionlimit(10**5)
from functools import lru_cache
#@lru_cache(None)
#======================================================#
input = lambda: sys.stdin.readline()
I = lambda: int(input().strip())
S = lambda: input().strip()
M = lambda: map(int,input().strip().split())
L = lambda: list(map(int,input().strip().split()))
#======================================================#
#======================================================#
def factorial_inverse():
mod=10**9+7
upto=(10**5)*3+1
nfactorial=[1 for i in range(upto)]
ninverse=[1 for i in range(upto)]
finverse=[1 for i in range(upto)]
for i in range(2,upto):
nfactorial[i]=(nfactorial[i-1]*i)%mod
ninverse[i]=(-(mod//i)*ninverse[mod%i])%mod
finverse[i]=(finverse[i-1]*ninverse[i])%mod
return nfactorial,finverse
def nCk(n,k):
if n<0 or k<0:
return 0
if k>n:
return 0
return ((nfactorial[n]*finverse[k]%mod)*finverse[n-k])%mod
#======================================================#
def primelist():
L = [False for i in range((10**10)+1)]
primes = [False for i in range((10**10)+1)]
for i in range(2,(10**10)+1):
if not L[i]:
primes[i]=True
for j in range(i,(10**10)+1,i):
L[j]=True
return primes
def isPrime(n):
p = primelist()
return p[n]
#======================================================#
def bst(arr,x):
low,high = 0,len(arr)-1
ans = -1
while low<=high:
mid = (low+high)//2
if arr[mid]==x:
return x
if arr[mid]<x:
low = mid+1
else:
high = mid-1
ans = arr[mid]
return ans
def factors(x):
l1 = []
l2 = []
for i in range(1,int(sqrt(x))+1):
if x%i==0:
if (i*i)==x:
l1.append(i)
else:
l1.append(i)
l2.append(x//i)
for i in range(len(l2)//2):
l2[i],l2[len(l2)-1-i]=l2[len(l2)-1-i],l2[i]
return l1+l2
#======================================================#
def power(n,x):
if x==0:
return 1
k = (10**9)+7
if n==1:
return 1
if x==1:
return n
ans = power(n,x//2)
if x%2==0:
return (ans*ans)%k
return (ans*ans*n)%k
#======================================================#
def f(x,y):
b = []
for i in range(k):
if x[i]==y[i]:
b.append(x[i])
else:
if x[i]=='S':
if y[i]=='E':
b.append('T')
else:
b.append('E')
elif x[i]=='E':
if y[i]=='S':
b.append('T')
else:
b.append('S')
else:
if y[i]=='S':
b.append('E')
else:
b.append('S')
return "".join(b)
n,k = M()
a = []
d = {}
ans = 0
for _ in range(n):
s = S()
a.append(s)
d[s]=0
for i in range(n-1):
for j in range(i+1,n):
if f(a[i],a[j]) in d:
ans+=1
print(ans//3)
| py |
1293 | B | B. JOE is on TV!time limit per test1 secondmemory limit per test256 megabytesinputstandard inputoutputstandard output3R2 - Standby for ActionOur dear Cafe's owner; JOE Miller, will soon take part in a new game TV-show "1 vs. nn"!The game goes in rounds, where in each round the host asks JOE and his opponents a common question. All participants failing to answer are eliminated. The show ends when only JOE remains (we assume that JOE never answers a question wrong!).For each question JOE answers, if there are ss (s>0s>0) opponents remaining and tt (0≤t≤s0≤t≤s) of them make a mistake on it, JOE receives tsts dollars, and consequently there will be s−ts−t opponents left for the next question.JOE wonders what is the maximum possible reward he can receive in the best possible scenario. Yet he has little time before show starts, so can you help him answering it instead?InputThe first and single line contains a single integer nn (1≤n≤1051≤n≤105), denoting the number of JOE's opponents in the show.OutputPrint a number denoting the maximum prize (in dollars) JOE could have.Your answer will be considered correct if it's absolute or relative error won't exceed 10−410−4. In other words, if your answer is aa and the jury answer is bb, then it must hold that |a−b|max(1,b)≤10−4|a−b|max(1,b)≤10−4.ExamplesInputCopy1
OutputCopy1.000000000000
InputCopy2
OutputCopy1.500000000000
NoteIn the second example; the best scenario would be: one contestant fails at the first question; the other fails at the next one. The total reward will be 12+11=1.512+11=1.5 dollars. | [
"combinatorics",
"greedy",
"math"
] | n = int(input())
s=float(1)
i=2
while i<=n:
s=s+(1/i)
i+=1
print(s) | py |
1320 | C | C. World of Darkraft: Battle for Azathothtime limit per test2 secondsmemory limit per test512 megabytesinputstandard inputoutputstandard outputRoma is playing a new expansion for his favorite game World of Darkraft. He made a new character and is going for his first grind.Roma has a choice to buy exactly one of nn different weapons and exactly one of mm different armor sets. Weapon ii has attack modifier aiai and is worth caicai coins, and armor set jj has defense modifier bjbj and is worth cbjcbj coins.After choosing his equipment Roma can proceed to defeat some monsters. There are pp monsters he can try to defeat. Monster kk has defense xkxk, attack ykyk and possesses zkzk coins. Roma can defeat a monster if his weapon's attack modifier is larger than the monster's defense, and his armor set's defense modifier is larger than the monster's attack. That is, a monster kk can be defeated with a weapon ii and an armor set jj if ai>xkai>xk and bj>ykbj>yk. After defeating the monster, Roma takes all the coins from them. During the grind, Roma can defeat as many monsters as he likes. Monsters do not respawn, thus each monster can be defeated at most one.Thanks to Roma's excessive donations, we can assume that he has an infinite amount of in-game currency and can afford any of the weapons and armor sets. Still, he wants to maximize the profit of the grind. The profit is defined as the total coins obtained from all defeated monsters minus the cost of his equipment. Note that Roma must purchase a weapon and an armor set even if he can not cover their cost with obtained coins.Help Roma find the maximum profit of the grind.InputThe first line contains three integers nn, mm, and pp (1≤n,m,p≤2⋅1051≤n,m,p≤2⋅105) — the number of available weapons, armor sets and monsters respectively.The following nn lines describe available weapons. The ii-th of these lines contains two integers aiai and caicai (1≤ai≤1061≤ai≤106, 1≤cai≤1091≤cai≤109) — the attack modifier and the cost of the weapon ii.The following mm lines describe available armor sets. The jj-th of these lines contains two integers bjbj and cbjcbj (1≤bj≤1061≤bj≤106, 1≤cbj≤1091≤cbj≤109) — the defense modifier and the cost of the armor set jj.The following pp lines describe monsters. The kk-th of these lines contains three integers xk,yk,zkxk,yk,zk (1≤xk,yk≤1061≤xk,yk≤106, 1≤zk≤1031≤zk≤103) — defense, attack and the number of coins of the monster kk.OutputPrint a single integer — the maximum profit of the grind.ExampleInputCopy2 3 3
2 3
4 7
2 4
3 2
5 11
1 2 4
2 1 6
3 4 6
OutputCopy1
| [
"brute force",
"data structures",
"sortings"
] | import sys, os, io
input = io.BytesIO(os.read(0, os.fstat(0).st_size)).readline
def update(l, r, s):
q, ll, rr, i = [1], [0], [l1 - 1], 0
while len(q) ^ i:
j = q[i]
l0, r0 = ll[i], rr[i]
if l <= l0 and r0 <= r:
lazy[j] += s
i += 1
continue
m0 = (l0 + r0) >> 1
if j < l1 and lazy[j]:
lazy[j << 1] += lazy[j]
lazy[j << 1 ^ 1] += lazy[j]
lazy[j] = 0
if l <= m0 and l0 <= r:
q.append(j << 1)
ll.append(l0)
rr.append(m0)
if l <= r0 and m0 + 1 <= r:
q.append(j << 1 ^ 1)
ll.append(m0 + 1)
rr.append(r0)
i += 1
for i in reversed(q):
if i < l1:
j, k = i << 1, i << 1 ^ 1
tree[i] = max(tree[j] + lazy[j], tree[k] + lazy[k])
return
n, m, p = map(int, input().split())
a = [tuple(map(int, input().split())) for _ in range(n)]
b = [tuple(map(int, input().split())) for _ in range(m)]
xyz = [tuple(map(int, input().split())) for _ in range(p)]
l = pow(10, 6) + 5
s = [0] * l
for b0, _ in b:
s[b0] = 1
for i in range(1, l):
s[i] += s[i - 1]
c = s[-1]
l1 = pow(2, (c + 5).bit_length())
l2 = 2 * l1
inf = 2 * pow(10, 9) + 1
tree, lazy = [-inf] * l2, [0] * l2
ma1 = -inf
for b0, cb in b:
ma1 = max(ma1, -cb)
tree[s[b0] + l1] = max(tree[s[b0] + l1], -cb)
for i in range(l1 - 1, 0, -1):
tree[i] = max(tree[2 * i], tree[2 * i + 1])
da = [-inf] * l
dx = [[] for _ in range(l)]
ma2 = -inf
for a0, ca in a:
ma2 = max(ma2, -ca)
da[a0] = max(da[a0], -ca)
for x, y, z in xyz:
dx[x].append((y, z))
ans = ma1 + ma2
for i in range(1, l):
if da[i] ^ -inf:
ans = max(ans, tree[1] + lazy[1] + da[i])
for y, z in dx[i]:
update(s[y] + 1, c + 1, z)
print(ans) | py |
1303 | A | A. Erasing Zeroestime limit per test1 secondmemory limit per test256 megabytesinputstandard inputoutputstandard outputYou are given a string ss. Each character is either 0 or 1.You want all 1's in the string to form a contiguous subsegment. For example, if the string is 0, 1, 00111 or 01111100, then all 1's form a contiguous subsegment, and if the string is 0101, 100001 or 11111111111101, then this condition is not met.You may erase some (possibly none) 0's from the string. What is the minimum number of 0's that you have to erase?InputThe first line contains one integer tt (1≤t≤1001≤t≤100) — the number of test cases.Then tt lines follow, each representing a test case. Each line contains one string ss (1≤|s|≤1001≤|s|≤100); each character of ss is either 0 or 1.OutputPrint tt integers, where the ii-th integer is the answer to the ii-th testcase (the minimum number of 0's that you have to erase from ss).ExampleInputCopy3
010011
0
1111000
OutputCopy2
0
0
NoteIn the first test case you have to delete the third and forth symbols from string 010011 (it turns into 0111). | [
"implementation",
"strings"
] | for _ in range(int(input())):
s=list(input())
zero=0
total=0
if '1' in s:
for i in s[s.index('1')::]:
if i=='0':
zero+=1
else:
total+=zero
zero=0
print(total)
| py |
1301 | A | A. Three Stringstime limit per test1 secondmemory limit per test256 megabytesinputstandard inputoutputstandard outputYou are given three strings aa, bb and cc of the same length nn. The strings consist of lowercase English letters only. The ii-th letter of aa is aiai, the ii-th letter of bb is bibi, the ii-th letter of cc is cici.For every ii (1≤i≤n1≤i≤n) you must swap (i.e. exchange) cici with either aiai or bibi. So in total you'll perform exactly nn swap operations, each of them either ci↔aici↔ai or ci↔bici↔bi (ii iterates over all integers between 11 and nn, inclusive).For example, if aa is "code", bb is "true", and cc is "help", you can make cc equal to "crue" taking the 11-st and the 44-th letters from aa and the others from bb. In this way aa becomes "hodp" and bb becomes "tele".Is it possible that after these swaps the string aa becomes exactly the same as the string bb?InputThe input consists of multiple test cases. The first line contains a single integer tt (1≤t≤1001≤t≤100) — the number of test cases. The description of the test cases follows.The first line of each test case contains a string of lowercase English letters aa.The second line of each test case contains a string of lowercase English letters bb.The third line of each test case contains a string of lowercase English letters cc.It is guaranteed that in each test case these three strings are non-empty and have the same length, which is not exceeding 100100.OutputPrint tt lines with answers for all test cases. For each test case:If it is possible to make string aa equal to string bb print "YES" (without quotes), otherwise print "NO" (without quotes).You can print either lowercase or uppercase letters in the answers.ExampleInputCopy4
aaa
bbb
ccc
abc
bca
bca
aabb
bbaa
baba
imi
mii
iim
OutputCopyNO
YES
YES
NO
NoteIn the first test case; it is impossible to do the swaps so that string aa becomes exactly the same as string bb.In the second test case, you should swap cici with aiai for all possible ii. After the swaps aa becomes "bca", bb becomes "bca" and cc becomes "abc". Here the strings aa and bb are equal.In the third test case, you should swap c1c1 with a1a1, c2c2 with b2b2, c3c3 with b3b3 and c4c4 with a4a4. Then string aa becomes "baba", string bb becomes "baba" and string cc becomes "abab". Here the strings aa and bb are equal.In the fourth test case, it is impossible to do the swaps so that string aa becomes exactly the same as string bb. | [
"implementation",
"strings"
] | t = int(input())
for i in range(t):
a = input()
b = input()
c = input()
for i in range(len(a)):
if c[i] != a[i] and c[i] != b[i]:
print("NO")
break
else:
print("YES") | py |
1296 | B | B. Food Buyingtime limit per test1 secondmemory limit per test256 megabytesinputstandard inputoutputstandard outputMishka wants to buy some food in the nearby shop. Initially; he has ss burles on his card. Mishka can perform the following operation any number of times (possibly, zero): choose some positive integer number 1≤x≤s1≤x≤s, buy food that costs exactly xx burles and obtain ⌊x10⌋⌊x10⌋ burles as a cashback (in other words, Mishka spends xx burles and obtains ⌊x10⌋⌊x10⌋ back). The operation ⌊ab⌋⌊ab⌋ means aa divided by bb rounded down.It is guaranteed that you can always buy some food that costs xx for any possible value of xx.Your task is to say the maximum number of burles Mishka can spend if he buys food optimally.For example, if Mishka has s=19s=19 burles then the maximum number of burles he can spend is 2121. Firstly, he can spend x=10x=10 burles, obtain 11 burle as a cashback. Now he has s=10s=10 burles, so can spend x=10x=10 burles, obtain 11 burle as a cashback and spend it too.You have to answer tt independent test cases.InputThe first line of the input contains one integer tt (1≤t≤1041≤t≤104) — the number of test cases.The next tt lines describe test cases. Each test case is given on a separate line and consists of one integer ss (1≤s≤1091≤s≤109) — the number of burles Mishka initially has.OutputFor each test case print the answer on it — the maximum number of burles Mishka can spend if he buys food optimally.ExampleInputCopy6
1
10
19
9876
12345
1000000000
OutputCopy1
11
21
10973
13716
1111111111
| [
"math"
] | t=int(input())
for r in range (t):
n=int(input())
spends=n
while n>=10:
spends+=n//10
n=n//10+n%10
print(spends) | py |
1307 | D | D. Cow and Fieldstime limit per test2 secondsmemory limit per test256 megabytesinputstandard inputoutputstandard outputBessie is out grazing on the farm; which consists of nn fields connected by mm bidirectional roads. She is currently at field 11, and will return to her home at field nn at the end of the day.The Cowfederation of Barns has ordered Farmer John to install one extra bidirectional road. The farm has kk special fields and he has decided to install the road between two different special fields. He may add the road between two special fields that already had a road directly connecting them.After the road is added, Bessie will return home on the shortest path from field 11 to field nn. Since Bessie needs more exercise, Farmer John must maximize the length of this shortest path. Help him!InputThe first line contains integers nn, mm, and kk (2≤n≤2⋅1052≤n≤2⋅105, n−1≤m≤2⋅105n−1≤m≤2⋅105, 2≤k≤n2≤k≤n) — the number of fields on the farm, the number of roads, and the number of special fields. The second line contains kk integers a1,a2,…,aka1,a2,…,ak (1≤ai≤n1≤ai≤n) — the special fields. All aiai are distinct.The ii-th of the following mm lines contains integers xixi and yiyi (1≤xi,yi≤n1≤xi,yi≤n, xi≠yixi≠yi), representing a bidirectional road between fields xixi and yiyi. It is guaranteed that one can reach any field from every other field. It is also guaranteed that for any pair of fields there is at most one road connecting them.OutputOutput one integer, the maximum possible length of the shortest path from field 11 to nn after Farmer John installs one road optimally.ExamplesInputCopy5 5 3
1 3 5
1 2
2 3
3 4
3 5
2 4
OutputCopy3
InputCopy5 4 2
2 4
1 2
2 3
3 4
4 5
OutputCopy3
NoteThe graph for the first example is shown below. The special fields are denoted by red. It is optimal for Farmer John to add a road between fields 33 and 55, and the resulting shortest path from 11 to 55 is length 33. The graph for the second example is shown below. Farmer John must add a road between fields 22 and 44, and the resulting shortest path from 11 to 55 is length 33. | [
"binary search",
"data structures",
"dfs and similar",
"graphs",
"greedy",
"shortest paths",
"sortings"
] | import os
import sys
from collections import defaultdict,deque
from io import BytesIO, IOBase
# MOD = 998244353
# nmax = 5000
# fact = [1] * (nmax+1)
# for i in range(2, nmax+1):
# fact[i] = fact[i-1] * i % MOD
# inv = [1] * (nmax+1)
# for i in range(2, nmax+1):
# inv[i] = pow(fact[i], MOD-2, MOD)
# def C(n, m):
# return fact[n] * inv[m] % MOD * inv[n-m] % MOD if 0 <= m <= n else 0
# from itertools import permutations
# from bisect import bisect_left
# d=[]
# for i in range(1,7):
# def solve(j,p):
# if j==i:
# return d.append(p)
# for el in [0,1]:
# solve(j+1,10*p+el)
# solve(0,0)
from collections import Counter
def main():
from collections import defaultdict, deque
def bfs(start, n):
dist=[-1]*n
q=deque()
visited=set()
q.append(start)
temp=0
while q:
n=len(q)
for _ in range(n):
curr=q.popleft()
if curr in visited:
continue
dist[curr]=temp
visited.add(curr)
for node in graph[curr]:
if node not in visited:
q.append(node)
temp+=1
return dist
n,m,k=map(int,input().split())
sp=list(map(int,input().split()))
sp=[el-1 for el in sp]
graph=defaultdict(list)
for _ in range(m):
a,b=map(int,input().split())
a-=1
b-=1
graph[a].append(b)
graph[b].append(a)
a=sp
dist_0=bfs(0,n)
dist_n=bfs(n-1,n)
ans=[]
for i in range(k):
ans.append((dist_0[a[i]]-dist_n[a[i]],a[i]))
ans.sort()
mx=[0]
for el in ans[::-1]:
mx.append(max(mx[-1],dist_n[el[1]]))
mx=mx[::-1]
res=0
for i in range(k-1):
curr=min(dist_0[n-1], 1+dist_0[ans[i][1]]+mx[i+1])
res=max(res,curr)
print(res)
#----------------------------------------------------------------------------------------
# region fastio
BUFSIZE = 8192
class FastIO(IOBase):
newlines = 0
def __init__(self, file):
self._fd = file.fileno()
self.buffer = BytesIO()
self.writable = 'x' in file.mode or 'r' not in file.mode
self.write = self.buffer.write if self.writable else None
def read(self):
while True:
b = os.read(self._fd, max(os.fstat(self._fd).st_size, BUFSIZE))
if not b:
break
ptr = self.buffer.tell()
self.buffer.seek(0, 2), self.buffer.write(b), self.buffer.seek(ptr)
self.newlines = 0
return self.buffer.read()
def readline(self):
while self.newlines == 0:
b = os.read(self._fd, max(os.fstat(self._fd).st_size, BUFSIZE))
self.newlines = b.count(b'\n') + (not b)
ptr = self.buffer.tell()
self.buffer.seek(0, 2), self.buffer.write(b), self.buffer.seek(ptr)
self.newlines -= 1
return self.buffer.readline()
def flush(self):
if self.writable:
os.write(self._fd, self.buffer.getvalue())
self.buffer.truncate(0), self.buffer.seek(0)
class IOWrapper(IOBase):
def __init__(self, file):
self.buffer = FastIO(file)
self.flush = self.buffer.flush
self.writable = self.buffer.writable
self.write = lambda s: self.buffer.write(s.encode('ascii'))
self.read = lambda: self.buffer.read().decode('ascii')
self.readline = lambda: self.buffer.readline().decode('ascii')
sys.stdin, sys.stdout = IOWrapper(sys.stdin), IOWrapper(sys.stdout)
input = lambda: sys.stdin.readline().rstrip('\r\n')
# endregion
if __name__ == '__main__':
main() | py |
1316 | A | A. Grade Allocationtime limit per test1 secondmemory limit per test256 megabytesinputstandard inputoutputstandard outputnn students are taking an exam. The highest possible score at this exam is mm. Let aiai be the score of the ii-th student. You have access to the school database which stores the results of all students.You can change each student's score as long as the following conditions are satisfied: All scores are integers 0≤ai≤m0≤ai≤m The average score of the class doesn't change. You are student 11 and you would like to maximize your own score.Find the highest possible score you can assign to yourself such that all conditions are satisfied.InputEach test contains multiple test cases. The first line contains the number of test cases tt (1≤t≤2001≤t≤200). The description of the test cases follows.The first line of each test case contains two integers nn and mm (1≤n≤1031≤n≤103, 1≤m≤1051≤m≤105) — the number of students and the highest possible score respectively.The second line of each testcase contains nn integers a1,a2,…,ana1,a2,…,an (0≤ai≤m0≤ai≤m) — scores of the students.OutputFor each testcase, output one integer — the highest possible score you can assign to yourself such that both conditions are satisfied._ExampleInputCopy2
4 10
1 2 3 4
4 5
1 2 3 4
OutputCopy10
5
NoteIn the first case; a=[1,2,3,4]a=[1,2,3,4], with average of 2.52.5. You can change array aa to [10,0,0,0][10,0,0,0]. Average remains 2.52.5, and all conditions are satisfied.In the second case, 0≤ai≤50≤ai≤5. You can change aa to [5,1,1,3][5,1,1,3]. You cannot increase a1a1 further as it will violate condition 0≤ai≤m0≤ai≤m. | [
"implementation"
] | def solve():
n,m = list(map(int,input().split()))
return min(sum(list(map(int,input().split()))),m)
for i in range(int(input())):
print(solve()) | py |
1296 | F | F. Berland Beautytime limit per test3 secondsmemory limit per test256 megabytesinputstandard inputoutputstandard outputThere are nn railway stations in Berland. They are connected to each other by n−1n−1 railway sections. The railway network is connected, i.e. can be represented as an undirected tree.You have a map of that network, so for each railway section you know which stations it connects.Each of the n−1n−1 sections has some integer value of the scenery beauty. However, these values are not marked on the map and you don't know them. All these values are from 11 to 106106 inclusive.You asked mm passengers some questions: the jj-th one told you three values: his departure station ajaj; his arrival station bjbj; minimum scenery beauty along the path from ajaj to bjbj (the train is moving along the shortest path from ajaj to bjbj). You are planning to update the map and set some value fifi on each railway section — the scenery beauty. The passengers' answers should be consistent with these values.Print any valid set of values f1,f2,…,fn−1f1,f2,…,fn−1, which the passengers' answer is consistent with or report that it doesn't exist.InputThe first line contains a single integer nn (2≤n≤50002≤n≤5000) — the number of railway stations in Berland.The next n−1n−1 lines contain descriptions of the railway sections: the ii-th section description is two integers xixi and yiyi (1≤xi,yi≤n,xi≠yi1≤xi,yi≤n,xi≠yi), where xixi and yiyi are the indices of the stations which are connected by the ii-th railway section. All the railway sections are bidirected. Each station can be reached from any other station by the railway.The next line contains a single integer mm (1≤m≤50001≤m≤5000) — the number of passengers which were asked questions. Then mm lines follow, the jj-th line contains three integers ajaj, bjbj and gjgj (1≤aj,bj≤n1≤aj,bj≤n; aj≠bjaj≠bj; 1≤gj≤1061≤gj≤106) — the departure station, the arrival station and the minimum scenery beauty along his path.OutputIf there is no answer then print a single integer -1.Otherwise, print n−1n−1 integers f1,f2,…,fn−1f1,f2,…,fn−1 (1≤fi≤1061≤fi≤106), where fifi is some valid scenery beauty along the ii-th railway section.If there are multiple answers, you can print any of them.ExamplesInputCopy4
1 2
3 2
3 4
2
1 2 5
1 3 3
OutputCopy5 3 5
InputCopy6
1 2
1 6
3 1
1 5
4 1
4
6 1 3
3 4 1
6 5 2
1 2 5
OutputCopy5 3 1 2 1
InputCopy6
1 2
1 6
3 1
1 5
4 1
4
6 1 1
3 4 3
6 5 3
1 2 4
OutputCopy-1
| [
"constructive algorithms",
"dfs and similar",
"greedy",
"sortings",
"trees"
] | import sys,math,itertools
from collections import Counter,deque,defaultdict
from bisect import bisect_left,bisect_right
from heapq import heappop,heappush,heapify
mod = 10**9+7
INF = float('inf')
def inp(): return int(sys.stdin.readline())
def inpl(): return list(map(int, sys.stdin.readline().split()))
def err():
print('-1'); exit()
n = inp()
g = [[] for _ in range(n)]
d = []
for i in range(n-1):
a,b = inpl()
a,b = a-1,b-1
g[a].append(b)
g[b].append(a)
if a>b: a,b = b,a
d.append([a,b])
bea = [[-1]*n for _ in range(n)]
cnt = [0] * (10**6+10)
m = inp()
pas = [inpl() for _ in range(m)]
pas.sort(key = lambda x:x[2])
root = []
for i,(a,b,c) in enumerate(pas):
a,b = a-1,b-1
pre = [-1] * n
q = deque([a])
f = True
while q and f:
nv = q.popleft()
for v in g[nv]:
if v == pre[nv]: continue
pre[v] = nv
if v == b:
ro = [v]
while v != a:
s,t = v,pre[v]
if s>t: s,t = t,s
bea[s][t] = c
ro.append(pre[v])
v = pre[v]
root.append(ro[::-1])
f = False
q.append(v)
for i,(a,b,c) in enumerate(pas):
a,b = a-1,b-1
bad = True
for j in range(len(root[i])-1):
s,t = root[i][j], root[i][j+1]
if s>t: s,t = t,s
if bea[s][t] == c: bad = False; break
if bad: err()
res = []
for i in range(n-1):
a,b = d[i]
now = bea[a][b]
if now == -1:
now = 1
res.append(now)
print(*res) | py |
1312 | C | C. Adding Powerstime limit per test2 secondsmemory limit per test256 megabytesinputstandard inputoutputstandard outputSuppose you are performing the following algorithm. There is an array v1,v2,…,vnv1,v2,…,vn filled with zeroes at start. The following operation is applied to the array several times — at ii-th step (00-indexed) you can: either choose position pospos (1≤pos≤n1≤pos≤n) and increase vposvpos by kiki; or not choose any position and skip this step. You can choose how the algorithm would behave on each step and when to stop it. The question is: can you make array vv equal to the given array aa (vj=ajvj=aj for each jj) after some step?InputThe first line contains one integer TT (1≤T≤10001≤T≤1000) — the number of test cases. Next 2T2T lines contain test cases — two lines per test case.The first line of each test case contains two integers nn and kk (1≤n≤301≤n≤30, 2≤k≤1002≤k≤100) — the size of arrays vv and aa and value kk used in the algorithm.The second line contains nn integers a1,a2,…,ana1,a2,…,an (0≤ai≤10160≤ai≤1016) — the array you'd like to achieve.OutputFor each test case print YES (case insensitive) if you can achieve the array aa after some step or NO (case insensitive) otherwise.ExampleInputCopy5
4 100
0 0 0 0
1 2
1
3 4
1 4 1
3 2
0 1 3
3 9
0 59049 810
OutputCopyYES
YES
NO
NO
YES
NoteIn the first test case; you can stop the algorithm before the 00-th step, or don't choose any position several times and stop the algorithm.In the second test case, you can add k0k0 to v1v1 and stop the algorithm.In the third test case, you can't make two 11 in the array vv.In the fifth test case, you can skip 9090 and 9191, then add 9292 and 9393 to v3v3, skip 9494 and finally, add 9595 to v2v2. | [
"bitmasks",
"greedy",
"implementation",
"math",
"number theory",
"ternary search"
] | # template taken from https://github.com/cheran-senthil/PyRival/blob/master/templates/template.py
import os
import sys
from io import BytesIO, IOBase
import math
from heapq import heappop, heappush, heapify, heapreplace
from collections import defaultdict, deque, OrderedDict
from bisect import bisect_left, bisect_right
#from functools import lru_cache
#import re
#from timeit import timeit
mod = 1000000007
mod1 = 998244353
#sys.setrecursionlimit(100090)
alpha = {'a': 0, 'b': 1, 'c': 2, 'd': 3, 'e': 4, 'f': 5, 'g': 6, 'h': 7, 'i': 8, 'j': 9, 'k': 10, 'l': 11, 'm': 12, 'n': 13, 'o': 14, 'p': 15, 'q': 16, 'r': 17, 's': 18, 't': 19, 'u': 20, 'v': 21, 'w': 22, 'x': 23, 'y': 24, 'z': 25}
def read():
sys.stdin = open('input.txt', 'r')
sys.stdout = open('output.txt', 'w')
# ------------------------------------------- Algos --------------------------------------------------------
def primeFactors(n):
arr = []
l = 0
while n % 2 == 0:
arr.append(2)
l += 1
n = n // 2
for i in range(3,int(math.sqrt(n))+1,2):
while n % i== 0:
arr.append(i)
l += 1
n = n // i
if n > 2:
arr.append(n)
l += 1
return arr
#return l
def primeFactorsSet(n):
s = set()
while n % 2 == 0:
s.add(2)
n = n // 2
for i in range(3,int(math.sqrt(n))+1,2):
while n % i== 0:
s.add(i)
n = n // i
if n > 2:
s.add(n)
return s
def sieve(n):
res=[0 for i in range(n+1)]
#prime=set([])
prime=[]
for i in range(2,n+1):
if not res[i]:
#prime.add(i)
prime.append(i)
for j in range(1,n//i+1):
res[i*j]=1
return prime
def isPrime(n) : # Check Prime Number or not
if (n <= 1) : return False
if (n <= 3) : return True
if (n % 2 == 0 or n % 3 == 0) : return False
i = 5
while(i * i <= n) :
if (n % i == 0 or n % (i + 2) == 0) :
return False
i = i + 6
return True
def countPrimesLessThanN(n):
seen, ans = [0] * n, 0
for num in range(2, n):
if seen[num]: continue
ans += 1
seen[num*num:n:num] = [1] * ((n - 1) // num - num + 1)
return ans
def gcd(x, y):
return math.gcd(x, y)
def lcm(a,b):
return (a // gcd(a,b))* b
def isBitSet(n,k):
# returns the count of numbers up to N having K-th bit set.
res = (n >> (k + 1)) << k
if ((n >> k) & 1):
res += n & ((1 << k) - 1)
return res
def popcount(x):
x = x - ((x >> 1) & 0x55555555)
x = (x & 0x33333333) + ((x >> 2) & 0x33333333)
x = (x + (x >> 4)) & 0x0f0f0f0f
x = x + (x >> 8)
x = x + (x >> 16)
return x & 0x0000007f
def modular_exponentiation(x, y, p): #returns (x^y)%p
res = 1
x = x % p
if (x == 0) :
return 0
while (y > 0) :
if ((y & 1) == 1) :
res = (res * x) % p
y = y >> 1
x = (x * x) % p
return res
def nCr(n, r, p): # returns nCr % p
num = den = 1
for i in range(r):
num = (num * (n - i)) % p
den = (den * (i + 1)) % p
return (num * pow(den,
p - 2, p)) % p
def smallestDivisor(n):
if (n % 2 == 0):
return 2
i = 3
while(i * i <= n):
if (n % i == 0):
return i
i += 2
return n
def numOfDivisors(n):
def pf(n):
dic = defaultdict(int)
while n % 2 == 0:
dic[2] += 1
n = n // 2
for i in range(3,int(math.sqrt(n))+1,2):
while n % i== 0:
dic[i] += 1
n = n // i
if n > 2:
dic[n] += 1
return dic
p = pf(n)
ans = 1
for item in p:
ans *= (p[item] + 1)
return ans
'''def SPF(spf, MAXN):
spf[1] = 1
for i in range(2, MAXN):
spf[i] = i
for i in range(4, MAXN, 2):
spf[i] = 2
for i in range(3, math.ceil(math.sqrt(MAXN))):
if (spf[i] == i):
for j in range(i * i, MAXN, i):
if (spf[j] == j):
spf[j] = i'''
# A O(log n) function returning prime
# factorization by dividing by smallest
# prime factor at every step
def factorizationInLogN(x, spf):
#ret = list()
dic = defaultdict(int)
while (x != 1):
#ret.append(spf[x])
dic[spf[x]] += 1
x = x // spf[x]
#return ret
return dic
'''MAXN = 10000001
spf = [0 for i in range(MAXN)]
SPF(spf, MAXN)'''
class DisjointSetUnion:
def __init__(self, n):
self.parent = list(range(n))
self.size = [1] * n
self.num_sets = n
def find(self, a):
acopy = a
while a != self.parent[a]:
a = self.parent[a]
while acopy != a:
self.parent[acopy], acopy = a, self.parent[acopy]
return a
def union(self, a, b):
a, b = self.find(a), self.find(b)
if a == b:
return False
if self.size[a] < self.size[b]:
a, b = b, a
self.num_sets -= 1
self.parent[b] = a
self.size[a] += self. size[b]
return True
def set_size(self, a):
return self.size[self.find(a)]
def __len__(self):
return self.num_sets
#a = sorted(range(n), key=arr.__getitem__)
# -------------------------------------------------- code -------------------------------------------------
def lis():
return [int(i) for i in input().split()]
def main():
t = int(input())
for _ in range(t):
n,k = map(int, input().split())
arr = lis()
mx = max(arr)
mask = []
pows = []
i = 1
while i <= mx:
pows.append(i)
mask.append(0)
i *= k
flag = True
for i in range(n):
while arr[i] > 0:
x = bisect_right(pows, arr[i])-1
if mask[x] == 1:
flag = False
break
mask[x] = 1
arr[i] -= pows[x]
if not flag:
break
if flag:
print("YES")
else:
print("NO")
# ---------------------------------------------- region fastio ---------------------------------------------
BUFSIZE = 8192
class FastIO(IOBase):
newlines = 0
def __init__(self, file):
self._fd = file.fileno()
self.buffer = BytesIO()
self.writable = "x" in file.mode or "r" not in file.mode
self.write = self.buffer.write if self.writable else None
def read(self):
while True:
b = os.read(self._fd, max(os.fstat(self._fd).st_size, BUFSIZE))
if not b:
break
ptr = self.buffer.tell()
self.buffer.seek(0, 2), self.buffer.write(b), self.buffer.seek(ptr)
self.newlines = 0
return self.buffer.read()
def readline(self):
while self.newlines == 0:
b = os.read(self._fd, max(os.fstat(self._fd).st_size, BUFSIZE))
self.newlines = b.count(b"\n") + (not b)
ptr = self.buffer.tell()
self.buffer.seek(0, 2), self.buffer.write(b), self.buffer.seek(ptr)
self.newlines -= 1
return self.buffer.readline()
def flush(self):
if self.writable:
os.write(self._fd, self.buffer.getvalue())
self.buffer.truncate(0), self.buffer.seek(0)
class IOWrapper(IOBase):
def __init__(self, file):
self.buffer = FastIO(file)
self.flush = self.buffer.flush
self.writable = self.buffer.writable
self.write = lambda s: self.buffer.write(s.encode("ascii"))
self.read = lambda: self.buffer.read().decode("ascii")
self.readline = lambda: self.buffer.readline().decode("ascii")
sys.stdin, sys.stdout = IOWrapper(sys.stdin), IOWrapper(sys.stdout)
input = lambda: sys.stdin.readline().rstrip("\r\n")
# -------------------------------------------------- end region ---------------------------------------------
if __name__ == "__main__":
#read()
main() | py |
1141 | B | B. Maximal Continuous Resttime limit per test2 secondsmemory limit per test256 megabytesinputstandard inputoutputstandard outputEach day in Berland consists of nn hours. Polycarp likes time management. That's why he has a fixed schedule for each day — it is a sequence a1,a2,…,ana1,a2,…,an (each aiai is either 00 or 11), where ai=0ai=0 if Polycarp works during the ii-th hour of the day and ai=1ai=1 if Polycarp rests during the ii-th hour of the day.Days go one after another endlessly and Polycarp uses the same schedule for each day.What is the maximal number of continuous hours during which Polycarp rests? It is guaranteed that there is at least one working hour in a day.InputThe first line contains nn (1≤n≤2⋅1051≤n≤2⋅105) — number of hours per day.The second line contains nn integer numbers a1,a2,…,ana1,a2,…,an (0≤ai≤10≤ai≤1), where ai=0ai=0 if the ii-th hour in a day is working and ai=1ai=1 if the ii-th hour is resting. It is guaranteed that ai=0ai=0 for at least one ii.OutputPrint the maximal number of continuous hours during which Polycarp rests. Remember that you should consider that days go one after another endlessly and Polycarp uses the same schedule for each day.ExamplesInputCopy5
1 0 1 0 1
OutputCopy2
InputCopy6
0 1 0 1 1 0
OutputCopy2
InputCopy7
1 0 1 1 1 0 1
OutputCopy3
InputCopy3
0 0 0
OutputCopy0
NoteIn the first example; the maximal rest starts in last hour and goes to the first hour of the next day.In the second example; Polycarp has maximal rest from the 44-th to the 55-th hour.In the third example, Polycarp has maximal rest from the 33-rd to the 55-th hour.In the fourth example, Polycarp has no rest at all. | [
"implementation"
] | n = int(input())
lst = [int(x) for x in input().split()][:n]
cnt, m_cnt = 0, 0
cnt2, m_cnt2 = 0, 0
pos = 0
if lst[0] == lst[-1] and lst[0] == 1:
for j in range(n - 1):
if lst[j] == 1:cnt += 1
else:
pos = j
break
if cnt > m_cnt: m_cnt = cnt
for j in range(n - 1, pos, -1):
if lst[j] == 1: cnt += 1
else: break
if cnt > m_cnt: m_cnt = cnt
for j in range(n):
if lst[j] == 1: cnt2 += 1
else: cnt2 = 0
if cnt2 > m_cnt2: m_cnt2 = cnt2
print(max(m_cnt, m_cnt2))
| py |
1294 | D | D. MEX maximizingtime limit per test3 secondsmemory limit per test256 megabytesinputstandard inputoutputstandard outputRecall that MEX of an array is a minimum non-negative integer that does not belong to the array. Examples: for the array [0,0,1,0,2][0,0,1,0,2] MEX equals to 33 because numbers 0,10,1 and 22 are presented in the array and 33 is the minimum non-negative integer not presented in the array; for the array [1,2,3,4][1,2,3,4] MEX equals to 00 because 00 is the minimum non-negative integer not presented in the array; for the array [0,1,4,3][0,1,4,3] MEX equals to 22 because 22 is the minimum non-negative integer not presented in the array. You are given an empty array a=[]a=[] (in other words, a zero-length array). You are also given a positive integer xx.You are also given qq queries. The jj-th query consists of one integer yjyj and means that you have to append one element yjyj to the array. The array length increases by 11 after a query.In one move, you can choose any index ii and set ai:=ai+xai:=ai+x or ai:=ai−xai:=ai−x (i.e. increase or decrease any element of the array by xx). The only restriction is that aiai cannot become negative. Since initially the array is empty, you can perform moves only after the first query.You have to maximize the MEX (minimum excluded) of the array if you can perform any number of such operations (you can even perform the operation multiple times with one element).You have to find the answer after each of qq queries (i.e. the jj-th answer corresponds to the array of length jj).Operations are discarded before each query. I.e. the array aa after the jj-th query equals to [y1,y2,…,yj][y1,y2,…,yj].InputThe first line of the input contains two integers q,xq,x (1≤q,x≤4⋅1051≤q,x≤4⋅105) — the number of queries and the value of xx.The next qq lines describe queries. The jj-th query consists of one integer yjyj (0≤yj≤1090≤yj≤109) and means that you have to append one element yjyj to the array.OutputPrint the answer to the initial problem after each query — for the query jj print the maximum value of MEX after first jj queries. Note that queries are dependent (the array changes after each query) but operations are independent between queries.ExamplesInputCopy7 3
0
1
2
2
0
0
10
OutputCopy1
2
3
3
4
4
7
InputCopy4 3
1
2
1
2
OutputCopy0
0
0
0
NoteIn the first example: After the first query; the array is a=[0]a=[0]: you don't need to perform any operations, maximum possible MEX is 11. After the second query, the array is a=[0,1]a=[0,1]: you don't need to perform any operations, maximum possible MEX is 22. After the third query, the array is a=[0,1,2]a=[0,1,2]: you don't need to perform any operations, maximum possible MEX is 33. After the fourth query, the array is a=[0,1,2,2]a=[0,1,2,2]: you don't need to perform any operations, maximum possible MEX is 33 (you can't make it greater with operations). After the fifth query, the array is a=[0,1,2,2,0]a=[0,1,2,2,0]: you can perform a[4]:=a[4]+3=3a[4]:=a[4]+3=3. The array changes to be a=[0,1,2,2,3]a=[0,1,2,2,3]. Now MEX is maximum possible and equals to 44. After the sixth query, the array is a=[0,1,2,2,0,0]a=[0,1,2,2,0,0]: you can perform a[4]:=a[4]+3=0+3=3a[4]:=a[4]+3=0+3=3. The array changes to be a=[0,1,2,2,3,0]a=[0,1,2,2,3,0]. Now MEX is maximum possible and equals to 44. After the seventh query, the array is a=[0,1,2,2,0,0,10]a=[0,1,2,2,0,0,10]. You can perform the following operations: a[3]:=a[3]+3=2+3=5a[3]:=a[3]+3=2+3=5, a[4]:=a[4]+3=0+3=3a[4]:=a[4]+3=0+3=3, a[5]:=a[5]+3=0+3=3a[5]:=a[5]+3=0+3=3, a[5]:=a[5]+3=3+3=6a[5]:=a[5]+3=3+3=6, a[6]:=a[6]−3=10−3=7a[6]:=a[6]−3=10−3=7, a[6]:=a[6]−3=7−3=4a[6]:=a[6]−3=7−3=4. The resulting array will be a=[0,1,2,5,3,6,4]a=[0,1,2,5,3,6,4]. Now MEX is maximum possible and equals to 77. | [
"data structures",
"greedy",
"implementation",
"math"
] | from bisect import *
from collections import *
import sys
import io, os
import math
import random
from heapq import *
gcd = math.gcd
sqrt = math.sqrt
maxint=10**21
def ceil(a, b):
if(b==0):
return maxint
a = -a
k = a // b
k = -k
return k
# arr=list(map(int, input().split()))
# n,m=map(int,input().split())
input = io.BytesIO(os.read(0, os.fstat(0).st_size)).readline
def strinp(testcases):
k = 5
if (testcases == -1 or testcases == 1):
k = 1
f = str(input())
f = f[2:len(f) - k]
return f
def lcm(n):
ans=1
for i in range(2,n):
ans=(ans*i)//(gcd(ans,i))
return(ans)
def main():
q,x=map(int,input().split())
ct=[0]*(q+1)
mex=0
if(q<x):
mex=0
for i in range(q):
g=int(input())
g=g%x
if(g>q):
print(mex)
continue
ct[g]+=1
while(ct[mex]):
mex+=1
print(mex)
return
groups=[0 for i in range(x)]
for i in range(q):
g=int(input())
g=g%x
ind=g+(groups[g]*x)
if(ind<=q):
ct[ind]+=1
groups[g]+=1
while(ct[mex]):
mex+=1
print(mex)
main() | py |
1305 | C | C. Kuroni and Impossible Calculationtime limit per test1 secondmemory limit per test256 megabytesinputstandard inputoutputstandard outputTo become the king of Codeforces; Kuroni has to solve the following problem.He is given nn numbers a1,a2,…,ana1,a2,…,an. Help Kuroni to calculate ∏1≤i<j≤n|ai−aj|∏1≤i<j≤n|ai−aj|. As result can be very big, output it modulo mm.If you are not familiar with short notation, ∏1≤i<j≤n|ai−aj|∏1≤i<j≤n|ai−aj| is equal to |a1−a2|⋅|a1−a3|⋅|a1−a2|⋅|a1−a3|⋅ …… ⋅|a1−an|⋅|a2−a3|⋅|a2−a4|⋅⋅|a1−an|⋅|a2−a3|⋅|a2−a4|⋅ …… ⋅|a2−an|⋅⋅|a2−an|⋅ …… ⋅|an−1−an|⋅|an−1−an|. In other words, this is the product of |ai−aj||ai−aj| for all 1≤i<j≤n1≤i<j≤n.InputThe first line contains two integers nn, mm (2≤n≤2⋅1052≤n≤2⋅105, 1≤m≤10001≤m≤1000) — number of numbers and modulo.The second line contains nn integers a1,a2,…,ana1,a2,…,an (0≤ai≤1090≤ai≤109).OutputOutput the single number — ∏1≤i<j≤n|ai−aj|modm∏1≤i<j≤n|ai−aj|modm.ExamplesInputCopy2 10
8 5
OutputCopy3InputCopy3 12
1 4 5
OutputCopy0InputCopy3 7
1 4 9
OutputCopy1NoteIn the first sample; |8−5|=3≡3mod10|8−5|=3≡3mod10.In the second sample, |1−4|⋅|1−5|⋅|4−5|=3⋅4⋅1=12≡0mod12|1−4|⋅|1−5|⋅|4−5|=3⋅4⋅1=12≡0mod12.In the third sample, |1−4|⋅|1−9|⋅|4−9|=3⋅8⋅5=120≡1mod7|1−4|⋅|1−9|⋅|4−9|=3⋅8⋅5=120≡1mod7. | [
"brute force",
"combinatorics",
"math",
"number theory"
] | n, m = map(int, input().split())
a = list(map(int, input().split()))
if n > m: exit(print(0))
ans = 1
for i in range(n):
for j in range(i+1, n):
ans *= abs(a[i] - a[j])
ans %= m
print(ans) | py |
1304 | B | B. Longest Palindrometime limit per test1 secondmemory limit per test256 megabytesinputstandard inputoutputstandard outputReturning back to problem solving; Gildong is now studying about palindromes. He learned that a palindrome is a string that is the same as its reverse. For example, strings "pop", "noon", "x", and "kkkkkk" are palindromes, while strings "moon", "tv", and "abab" are not. An empty string is also a palindrome.Gildong loves this concept so much, so he wants to play with it. He has nn distinct strings of equal length mm. He wants to discard some of the strings (possibly none or all) and reorder the remaining strings so that the concatenation becomes a palindrome. He also wants the palindrome to be as long as possible. Please help him find one.InputThe first line contains two integers nn and mm (1≤n≤1001≤n≤100, 1≤m≤501≤m≤50) — the number of strings and the length of each string.Next nn lines contain a string of length mm each, consisting of lowercase Latin letters only. All strings are distinct.OutputIn the first line, print the length of the longest palindrome string you made.In the second line, print that palindrome. If there are multiple answers, print any one of them. If the palindrome is empty, print an empty line or don't print this line at all.ExamplesInputCopy3 3
tab
one
bat
OutputCopy6
tabbat
InputCopy4 2
oo
ox
xo
xx
OutputCopy6
oxxxxo
InputCopy3 5
hello
codef
orces
OutputCopy0
InputCopy9 4
abab
baba
abcd
bcde
cdef
defg
wxyz
zyxw
ijji
OutputCopy20
ababwxyzijjizyxwbaba
NoteIn the first example; "battab" is also a valid answer.In the second example; there can be 4 different valid answers including the sample output. We are not going to provide any hints for what the others are.In the third example; the empty string is the only valid palindrome string. | [
"brute force",
"constructive algorithms",
"greedy",
"implementation",
"strings"
] | s = input().split(' ')
n = int(s[0])
m1 = int(s[1]) - 1
m = m1
l = []
s = ""
def f(str1):
i1 = m + 1
i2 = m
while i1 != 0 and i1 != 1:
if str1[i2] != str1[m - i2]:
return False
else:
i2 -= 1
i1 -= 2
return True
for i in range(n):
obj = input()
if f(obj):
s = obj
else:
l.append(obj)
while len(l) != 0:
for obj1 in l:
l.remove(obj1)
s1 = obj1[0]
for obj2 in l:
m = m1
s2 = obj2[m]
if s1 == s2:
r = obj1 + obj2
m = len(r) - 1
if f(r):
s = obj1 + s + obj2
l.remove(obj2)
print(len(s))
print(s)
| py |
1305 | B | B. Kuroni and Simple Stringstime limit per test1 secondmemory limit per test256 megabytesinputstandard inputoutputstandard outputNow that Kuroni has reached 10 years old; he is a big boy and doesn't like arrays of integers as presents anymore. This year he wants a Bracket sequence as a Birthday present. More specifically, he wants a bracket sequence so complex that no matter how hard he tries, he will not be able to remove a simple subsequence!We say that a string formed by nn characters '(' or ')' is simple if its length nn is even and positive, its first n2n2 characters are '(', and its last n2n2 characters are ')'. For example, the strings () and (()) are simple, while the strings )( and ()() are not simple.Kuroni will be given a string formed by characters '(' and ')' (the given string is not necessarily simple). An operation consists of choosing a subsequence of the characters of the string that forms a simple string and removing all the characters of this subsequence from the string. Note that this subsequence doesn't have to be continuous. For example, he can apply the operation to the string ')()(()))', to choose a subsequence of bold characters, as it forms a simple string '(())', delete these bold characters from the string and to get '))()'. Kuroni has to perform the minimum possible number of operations on the string, in such a way that no more operations can be performed on the remaining string. The resulting string does not have to be empty.Since the given string is too large, Kuroni is unable to figure out how to minimize the number of operations. Can you help him do it instead?A sequence of characters aa is a subsequence of a string bb if aa can be obtained from bb by deletion of several (possibly, zero or all) characters.InputThe only line of input contains a string ss (1≤|s|≤10001≤|s|≤1000) formed by characters '(' and ')', where |s||s| is the length of ss.OutputIn the first line, print an integer kk — the minimum number of operations you have to apply. Then, print 2k2k lines describing the operations in the following format:For each operation, print a line containing an integer mm — the number of characters in the subsequence you will remove.Then, print a line containing mm integers 1≤a1<a2<⋯<am1≤a1<a2<⋯<am — the indices of the characters you will remove. All integers must be less than or equal to the length of the current string, and the corresponding subsequence must form a simple string.If there are multiple valid sequences of operations with the smallest kk, you may print any of them.ExamplesInputCopy(()((
OutputCopy1
2
1 3
InputCopy)(
OutputCopy0
InputCopy(()())
OutputCopy1
4
1 2 5 6
NoteIn the first sample; the string is '(()(('. The operation described corresponds to deleting the bolded subsequence. The resulting string is '((('; and no more operations can be performed on it. Another valid answer is choosing indices 22 and 33, which results in the same final string.In the second sample, it is already impossible to perform any operations. | [
"constructive algorithms",
"greedy",
"strings",
"two pointers"
] | s = input()
r = set()
def foo():
global s, r
rem = []
p1, p2 = 0, len(s)-1
op = True
while p1 <= p2:
if op:
if s[p1] == '(' and p1 not in r:
rem.append(p1)
op = False
p1 += 1
else:
if s[p2] == ')' and p2 not in r:
rem.append(p2)
op = True
p2 -= 1
if not op: rem.pop()
n = len(rem)
r.update(rem)
rem = ' '.join(map(lambda x : str(x+1), sorted(rem)))
return n, rem
res = []
while True:
n, rem = foo()
if not n: break
res.append((n, rem))
print(len(res))
for n, rem in res:
print(n)
print(rem)
| py |
1299 | C | C. Water Balancetime limit per test3 secondsmemory limit per test256 megabytesinputstandard inputoutputstandard outputThere are nn water tanks in a row, ii-th of them contains aiai liters of water. The tanks are numbered from 11 to nn from left to right.You can perform the following operation: choose some subsegment [l,r][l,r] (1≤l≤r≤n1≤l≤r≤n), and redistribute water in tanks l,l+1,…,rl,l+1,…,r evenly. In other words, replace each of al,al+1,…,aral,al+1,…,ar by al+al+1+⋯+arr−l+1al+al+1+⋯+arr−l+1. For example, if for volumes [1,3,6,7][1,3,6,7] you choose l=2,r=3l=2,r=3, new volumes of water will be [1,4.5,4.5,7][1,4.5,4.5,7]. You can perform this operation any number of times.What is the lexicographically smallest sequence of volumes of water that you can achieve?As a reminder:A sequence aa is lexicographically smaller than a sequence bb of the same length if and only if the following holds: in the first (leftmost) position where aa and bb differ, the sequence aa has a smaller element than the corresponding element in bb.InputThe first line contains an integer nn (1≤n≤1061≤n≤106) — the number of water tanks.The second line contains nn integers a1,a2,…,ana1,a2,…,an (1≤ai≤1061≤ai≤106) — initial volumes of water in the water tanks, in liters.Because of large input, reading input as doubles is not recommended.OutputPrint the lexicographically smallest sequence you can get. In the ii-th line print the final volume of water in the ii-th tank.Your answer is considered correct if the absolute or relative error of each aiai does not exceed 10−910−9.Formally, let your answer be a1,a2,…,ana1,a2,…,an, and the jury's answer be b1,b2,…,bnb1,b2,…,bn. Your answer is accepted if and only if |ai−bi|max(1,|bi|)≤10−9|ai−bi|max(1,|bi|)≤10−9 for each ii.ExamplesInputCopy4
7 5 5 7
OutputCopy5.666666667
5.666666667
5.666666667
7.000000000
InputCopy5
7 8 8 10 12
OutputCopy7.000000000
8.000000000
8.000000000
10.000000000
12.000000000
InputCopy10
3 9 5 5 1 7 5 3 8 7
OutputCopy3.000000000
5.000000000
5.000000000
5.000000000
5.000000000
5.000000000
5.000000000
5.000000000
7.500000000
7.500000000
NoteIn the first sample; you can get the sequence by applying the operation for subsegment [1,3][1,3].In the second sample, you can't get any lexicographically smaller sequence. | [
"data structures",
"geometry",
"greedy"
] | from __future__ import division, print_function
import io
import os
input = io.BytesIO(os.read(0, os.fstat(0).st_size)).readline
n = int(input())
l = list(map(int, input().split()))
# not my solution, from: https://codeforces.com/contest/1299/submission/70653333
su=[l[0]]
cou=[-1,0]
for k in range(1,n):
nd=1
ns=l[k]
while len(cou)>1 and su[-1]*(cou[-1]-cou[-2]+nd)>(su[-1]+ns)*(cou[-1]-cou[-2]):
nd+=cou[-1]-cou[-2]
ns+=su[-1]
su.pop()
cou.pop()
cou.append(k)
su.append(ns)
floats = []
for k in range(len(su)):
floats += [su[k] / (cou[k + 1] - cou[k])] * (cou[k + 1] - cou[k])
if False:
if False:
# 3244 ms, TLE on pypy3
# But 1044 ms on pypy2!!!
print("\n".join(map(str, floats)))
if False:
# 2028 ms
os.write(1, b"\n".join(str(x).encode() for x in floats))
if False:
# 1544 ms
# %-formatting for bytes, added in python 3.5: https://www.python.org/dev/peps/pep-0461/
os.write(1, b"\n".join(b"%.9f" % x for x in floats))
if False:
# 842 ms
# Format with ints instead. TODO: handle negatives
os.write(1, b"\n".join(b"%d.%09d" % (x, (x - int(x)) * 1e9 + 0.5) for x in floats))
if True:
# 748 ms
# Same as above
ans = bytearray()
for x in floats:
i = int(x)
ans += b"%d.%09d\n" % (i, (x - i) * 1e9 + 0.5)
os.write(1, ans)
else:
# 670 ms
# Use that fact that most of the output are repeats (specific to this problem)
ans = bytearray()
for k in range(len(su)):
repeat = cou[k + 1] - cou[k]
x = su[k] / repeat
i = int(x)
ans += (b"%d.%09d\n" % (i, (x - i) * 1e9 + 0.5)) * repeat
os.write(1, ans)
| py |
1303 | E | E. Erase Subsequencestime limit per test2 secondsmemory limit per test256 megabytesinputstandard inputoutputstandard outputYou are given a string ss. You can build new string pp from ss using the following operation no more than two times: choose any subsequence si1,si2,…,siksi1,si2,…,sik where 1≤i1<i2<⋯<ik≤|s|1≤i1<i2<⋯<ik≤|s|; erase the chosen subsequence from ss (ss can become empty); concatenate chosen subsequence to the right of the string pp (in other words, p=p+si1si2…sikp=p+si1si2…sik). Of course, initially the string pp is empty. For example, let s=ababcds=ababcd. At first, let's choose subsequence s1s4s5=abcs1s4s5=abc — we will get s=bads=bad and p=abcp=abc. At second, let's choose s1s2=bas1s2=ba — we will get s=ds=d and p=abcbap=abcba. So we can build abcbaabcba from ababcdababcd.Can you build a given string tt using the algorithm above?InputThe first line contains the single integer TT (1≤T≤1001≤T≤100) — the number of test cases.Next 2T2T lines contain test cases — two per test case. The first line contains string ss consisting of lowercase Latin letters (1≤|s|≤4001≤|s|≤400) — the initial string.The second line contains string tt consisting of lowercase Latin letters (1≤|t|≤|s|1≤|t|≤|s|) — the string you'd like to build.It's guaranteed that the total length of strings ss doesn't exceed 400400.OutputPrint TT answers — one per test case. Print YES (case insensitive) if it's possible to build tt and NO (case insensitive) otherwise.ExampleInputCopy4
ababcd
abcba
a
b
defi
fed
xyz
x
OutputCopyYES
NO
NO
YES
| [
"dp",
"strings"
] | tt=int(input())
for _ in range(tt):
s=input()
t=input()
flag='NO'
j=0
ptr=0
while(j<len(s) and ptr<len(t)):
if(s[j]==t[ptr]):
ptr+=1
j+=1
else:
j+=1
if(ptr==len(t)):
flag='YES'
else:
pos=[0]*26
for i in range(len(s)):
pos[ord(s[i])-97]+=1
for i in range(0,len(t)):
h=[]
for j in range(0,len(pos)):
h.append(pos[j])
j=0
ptr=0
temp1=0
while(ptr<=i and j<len(s)):
if(s[j]==t[ptr] and h[ord(s[j])-97]>0):
h[ord(s[j])-97]-=1
ptr+=1
j+=1
else:
j+=1
if(ptr==i+1):
temp1=1
j=0
ptr=i+1
temp2=0
while(ptr<len(t) and j<len(s)):
if(s[j]==t[ptr] and h[ord(s[j])-97]>0):
h[ord(s[j])-97]-=1
ptr+=1
j+=1
else:
j+=1
if(ptr==len(t)):
temp2=1
if(temp1==1 and temp2==1):
flag='YES'
break
if(len(t)>105 and (t[:106]=='deabbaaeaceeadfafecfddcabcaabcbfeecfcceaecbaedebbffdcacbadafeeeaededcadeafdccadadeccdadefcbcdabcbeebbbbfae' or t[:106]=='dfbcaefcfcdecffeddaebfbacdefcbafdebdcdaebaecfdadcacfeddcfddaffdacfcfcfdaefcfaeadefededdeffdffcabeafeecabab')):
flag='NO'
print(flag)
| py |
1284 | D | D. New Year and Conferencetime limit per test2 secondsmemory limit per test1024 megabytesinputstandard inputoutputstandard outputFilled with optimism; Hyunuk will host a conference about how great this new year will be!The conference will have nn lectures. Hyunuk has two candidate venues aa and bb. For each of the nn lectures, the speaker specified two time intervals [sai,eai][sai,eai] (sai≤eaisai≤eai) and [sbi,ebi][sbi,ebi] (sbi≤ebisbi≤ebi). If the conference is situated in venue aa, the lecture will be held from saisai to eaieai, and if the conference is situated in venue bb, the lecture will be held from sbisbi to ebiebi. Hyunuk will choose one of these venues and all lectures will be held at that venue.Two lectures are said to overlap if they share any point in time in common. Formally, a lecture held in interval [x,y][x,y] overlaps with a lecture held in interval [u,v][u,v] if and only if max(x,u)≤min(y,v)max(x,u)≤min(y,v).We say that a participant can attend a subset ss of the lectures if the lectures in ss do not pairwise overlap (i.e. no two lectures overlap). Note that the possibility of attending may depend on whether Hyunuk selected venue aa or venue bb to hold the conference.A subset of lectures ss is said to be venue-sensitive if, for one of the venues, the participant can attend ss, but for the other venue, the participant cannot attend ss.A venue-sensitive set is problematic for a participant who is interested in attending the lectures in ss because the participant cannot be sure whether the lecture times will overlap. Hyunuk will be happy if and only if there are no venue-sensitive sets. Determine whether Hyunuk will be happy.InputThe first line contains an integer nn (1≤n≤1000001≤n≤100000), the number of lectures held in the conference.Each of the next nn lines contains four integers saisai, eaieai, sbisbi, ebiebi (1≤sai,eai,sbi,ebi≤1091≤sai,eai,sbi,ebi≤109, sai≤eai,sbi≤ebisai≤eai,sbi≤ebi).OutputPrint "YES" if Hyunuk will be happy. Print "NO" otherwise.You can print each letter in any case (upper or lower).ExamplesInputCopy2
1 2 3 6
3 4 7 8
OutputCopyYES
InputCopy3
1 3 2 4
4 5 6 7
3 4 5 5
OutputCopyNO
InputCopy6
1 5 2 9
2 4 5 8
3 6 7 11
7 10 12 16
8 11 13 17
9 12 14 18
OutputCopyYES
NoteIn second example; lecture set {1,3}{1,3} is venue-sensitive. Because participant can't attend this lectures in venue aa, but can attend in venue bb.In first and third example, venue-sensitive set does not exist. | [
"binary search",
"data structures",
"hashing",
"sortings"
] | MOD = 10**18+13
import sys
readline = sys.stdin.readline
from random import randrange
def compress(L):
L2 = list(set(L))
L2.sort()
C = {v : k for k, v in enumerate(L2, 1)}
return L2, C
N = int(readline())
T = set()
L = [None]*N
for i in range(N):
L[i] = tuple(map(int, readline().split()))
T |= set(L[i])
t2, C = compress(list(T))
Z = len(t2)
L = [[j] + [C[L[j][i]] for i in range(4)] for j in range(N)]
seed = []
while len(set(seed)) != N:
seed = [randrange(1, MOD) for _ in range(N)]
S1 = [0]*(Z+2)
S2 = [0]*(Z+2)
S3 = [0]*(Z+2)
S4 = [0]*(Z+2)
p1 = [0]*N
p2 = [0]*N
for i in range(N):
j, s1, t1, s2, t2 = L[i]
S1[s1] += seed[j]
S2[t1] += seed[j]
S3[s2] += seed[j]
S4[t2] += seed[j]
for i in range(1, Z+1):
S1[i] = (S1[i] + S1[i-1])%MOD
S2[i] = (S2[i] + S2[i-1])%MOD
S3[i] = (S3[i] + S3[i-1])%MOD
S4[i] = (S4[i] + S4[i-1])%MOD
for i in range(N):
j, s1, t1, s2, t2 = L[i]
p1[j] = (S2[s1-1] - S1[t1])%MOD
p2[j] = (S4[s2-1] - S3[t2])%MOD
print('YES' if all(a == b for a, b in zip(p1, p2)) else 'NO')
| py |
1323 | B | B. Count Subrectanglestime limit per test1 secondmemory limit per test512 megabytesinputstandard inputoutputstandard outputYou are given an array aa of length nn and array bb of length mm both consisting of only integers 00 and 11. Consider a matrix cc of size n×mn×m formed by following rule: ci,j=ai⋅bjci,j=ai⋅bj (i.e. aiai multiplied by bjbj). It's easy to see that cc consists of only zeroes and ones too.How many subrectangles of size (area) kk consisting only of ones are there in cc?A subrectangle is an intersection of a consecutive (subsequent) segment of rows and a consecutive (subsequent) segment of columns. I.e. consider four integers x1,x2,y1,y2x1,x2,y1,y2 (1≤x1≤x2≤n1≤x1≤x2≤n, 1≤y1≤y2≤m1≤y1≤y2≤m) a subrectangle c[x1…x2][y1…y2]c[x1…x2][y1…y2] is an intersection of the rows x1,x1+1,x1+2,…,x2x1,x1+1,x1+2,…,x2 and the columns y1,y1+1,y1+2,…,y2y1,y1+1,y1+2,…,y2.The size (area) of a subrectangle is the total number of cells in it.InputThe first line contains three integers nn, mm and kk (1≤n,m≤40000,1≤k≤n⋅m1≤n,m≤40000,1≤k≤n⋅m), length of array aa, length of array bb and required size of subrectangles.The second line contains nn integers a1,a2,…,ana1,a2,…,an (0≤ai≤10≤ai≤1), elements of aa.The third line contains mm integers b1,b2,…,bmb1,b2,…,bm (0≤bi≤10≤bi≤1), elements of bb.OutputOutput single integer — the number of subrectangles of cc with size (area) kk consisting only of ones.ExamplesInputCopy3 3 2
1 0 1
1 1 1
OutputCopy4
InputCopy3 5 4
1 1 1
1 1 1 1 1
OutputCopy14
NoteIn first example matrix cc is: There are 44 subrectangles of size 22 consisting of only ones in it: In second example matrix cc is: | [
"binary search",
"greedy",
"implementation"
] | import os
import sys
from io import BytesIO, IOBase
from collections import Counter, defaultdict
from sys import stdin, stdout
import io
from math import *
import heapq
import bisect
import collections
def ceil(a, b):
return (a + b - 1) // b
inf = float('inf')
def get():
return stdin.readline().rstrip()
mod = 10 ** 5 + 7
# for _ in range(int(get())):
# n=int(get())
# l=list(map(int,get().split()))
# = map(int,get().split())
#####################################################################
def divisors(n):
i = 1
l=set()
while (i * i < n):
if (n % i == 0):
l.add(i)
i += 1
for i in range(int(sqrt(n)), 0, -1):
if (n % i == 0):
l.add(n//i)
return l
#####################################################################
n,m,y = map(int,get().split())
l1=list(map(int,get().split()))
l2=list(map(int,get().split()))
l=divisors(y)
i=0
s1=defaultdict(int)
s2=defaultdict(int)
while i<n:
x=0
while i<n and l1[i]==1:
x+=1
i+=1
i+=1
s1[x]+=1
i=0
while i<m:
x=0
while i<m and l2[i]==1:
x+=1
i+=1
i+=1
s2[x] += 1
s3=[]
ans=0
# print(s1,s2)
for i in s1:
for j in s2:
for k in l:
if y%k==0:
if k<=i and y//k<=j:
ans+=(i-k+1)*(j-y//k+1)*s1[i]*s2[j]
print(ans)
| py |
1294 | D | D. MEX maximizingtime limit per test3 secondsmemory limit per test256 megabytesinputstandard inputoutputstandard outputRecall that MEX of an array is a minimum non-negative integer that does not belong to the array. Examples: for the array [0,0,1,0,2][0,0,1,0,2] MEX equals to 33 because numbers 0,10,1 and 22 are presented in the array and 33 is the minimum non-negative integer not presented in the array; for the array [1,2,3,4][1,2,3,4] MEX equals to 00 because 00 is the minimum non-negative integer not presented in the array; for the array [0,1,4,3][0,1,4,3] MEX equals to 22 because 22 is the minimum non-negative integer not presented in the array. You are given an empty array a=[]a=[] (in other words, a zero-length array). You are also given a positive integer xx.You are also given qq queries. The jj-th query consists of one integer yjyj and means that you have to append one element yjyj to the array. The array length increases by 11 after a query.In one move, you can choose any index ii and set ai:=ai+xai:=ai+x or ai:=ai−xai:=ai−x (i.e. increase or decrease any element of the array by xx). The only restriction is that aiai cannot become negative. Since initially the array is empty, you can perform moves only after the first query.You have to maximize the MEX (minimum excluded) of the array if you can perform any number of such operations (you can even perform the operation multiple times with one element).You have to find the answer after each of qq queries (i.e. the jj-th answer corresponds to the array of length jj).Operations are discarded before each query. I.e. the array aa after the jj-th query equals to [y1,y2,…,yj][y1,y2,…,yj].InputThe first line of the input contains two integers q,xq,x (1≤q,x≤4⋅1051≤q,x≤4⋅105) — the number of queries and the value of xx.The next qq lines describe queries. The jj-th query consists of one integer yjyj (0≤yj≤1090≤yj≤109) and means that you have to append one element yjyj to the array.OutputPrint the answer to the initial problem after each query — for the query jj print the maximum value of MEX after first jj queries. Note that queries are dependent (the array changes after each query) but operations are independent between queries.ExamplesInputCopy7 3
0
1
2
2
0
0
10
OutputCopy1
2
3
3
4
4
7
InputCopy4 3
1
2
1
2
OutputCopy0
0
0
0
NoteIn the first example: After the first query; the array is a=[0]a=[0]: you don't need to perform any operations, maximum possible MEX is 11. After the second query, the array is a=[0,1]a=[0,1]: you don't need to perform any operations, maximum possible MEX is 22. After the third query, the array is a=[0,1,2]a=[0,1,2]: you don't need to perform any operations, maximum possible MEX is 33. After the fourth query, the array is a=[0,1,2,2]a=[0,1,2,2]: you don't need to perform any operations, maximum possible MEX is 33 (you can't make it greater with operations). After the fifth query, the array is a=[0,1,2,2,0]a=[0,1,2,2,0]: you can perform a[4]:=a[4]+3=3a[4]:=a[4]+3=3. The array changes to be a=[0,1,2,2,3]a=[0,1,2,2,3]. Now MEX is maximum possible and equals to 44. After the sixth query, the array is a=[0,1,2,2,0,0]a=[0,1,2,2,0,0]: you can perform a[4]:=a[4]+3=0+3=3a[4]:=a[4]+3=0+3=3. The array changes to be a=[0,1,2,2,3,0]a=[0,1,2,2,3,0]. Now MEX is maximum possible and equals to 44. After the seventh query, the array is a=[0,1,2,2,0,0,10]a=[0,1,2,2,0,0,10]. You can perform the following operations: a[3]:=a[3]+3=2+3=5a[3]:=a[3]+3=2+3=5, a[4]:=a[4]+3=0+3=3a[4]:=a[4]+3=0+3=3, a[5]:=a[5]+3=0+3=3a[5]:=a[5]+3=0+3=3, a[5]:=a[5]+3=3+3=6a[5]:=a[5]+3=3+3=6, a[6]:=a[6]−3=10−3=7a[6]:=a[6]−3=10−3=7, a[6]:=a[6]−3=7−3=4a[6]:=a[6]−3=7−3=4. The resulting array will be a=[0,1,2,5,3,6,4]a=[0,1,2,5,3,6,4]. Now MEX is maximum possible and equals to 77. | [
"data structures",
"greedy",
"implementation",
"math"
] | import io, os
input = io.BytesIO(os.read(0,os.fstat(0).st_size)).readline
q,x=map(int,input().split())
cant = [0]*(x+5)
res=0
for i in range (q):
val = int (input())
val %=x
cant[val]+=1
while cant[res%x]>0:
cant[res%x]-=1
res+=1
print(res) | py |
1299 | A | A. Anu Has a Functiontime limit per test1 secondmemory limit per test256 megabytesinputstandard inputoutputstandard outputAnu has created her own function ff: f(x,y)=(x|y)−yf(x,y)=(x|y)−y where || denotes the bitwise OR operation. For example, f(11,6)=(11|6)−6=15−6=9f(11,6)=(11|6)−6=15−6=9. It can be proved that for any nonnegative numbers xx and yy value of f(x,y)f(x,y) is also nonnegative. She would like to research more about this function and has created multiple problems for herself. But she isn't able to solve all of them and needs your help. Here is one of these problems.A value of an array [a1,a2,…,an][a1,a2,…,an] is defined as f(f(…f(f(a1,a2),a3),…an−1),an)f(f(…f(f(a1,a2),a3),…an−1),an) (see notes). You are given an array with not necessarily distinct elements. How should you reorder its elements so that the value of the array is maximal possible?InputThe first line contains a single integer nn (1≤n≤1051≤n≤105).The second line contains nn integers a1,a2,…,ana1,a2,…,an (0≤ai≤1090≤ai≤109). Elements of the array are not guaranteed to be different.OutputOutput nn integers, the reordering of the array with maximum value. If there are multiple answers, print any.ExamplesInputCopy4
4 0 11 6
OutputCopy11 6 4 0InputCopy1
13
OutputCopy13 NoteIn the first testcase; value of the array [11,6,4,0][11,6,4,0] is f(f(f(11,6),4),0)=f(f(9,4),0)=f(9,0)=9f(f(f(11,6),4),0)=f(f(9,4),0)=f(9,0)=9.[11,4,0,6][11,4,0,6] is also a valid answer. | [
"brute force",
"greedy",
"math"
] | input()
a=input().split()
print(a.pop(next((x for x in zip(*(f'{int(x):30b}'for
x in a))if x.count('1')==1),'1').index('1')),*a)
| py |
1301 | A | A. Three Stringstime limit per test1 secondmemory limit per test256 megabytesinputstandard inputoutputstandard outputYou are given three strings aa, bb and cc of the same length nn. The strings consist of lowercase English letters only. The ii-th letter of aa is aiai, the ii-th letter of bb is bibi, the ii-th letter of cc is cici.For every ii (1≤i≤n1≤i≤n) you must swap (i.e. exchange) cici with either aiai or bibi. So in total you'll perform exactly nn swap operations, each of them either ci↔aici↔ai or ci↔bici↔bi (ii iterates over all integers between 11 and nn, inclusive).For example, if aa is "code", bb is "true", and cc is "help", you can make cc equal to "crue" taking the 11-st and the 44-th letters from aa and the others from bb. In this way aa becomes "hodp" and bb becomes "tele".Is it possible that after these swaps the string aa becomes exactly the same as the string bb?InputThe input consists of multiple test cases. The first line contains a single integer tt (1≤t≤1001≤t≤100) — the number of test cases. The description of the test cases follows.The first line of each test case contains a string of lowercase English letters aa.The second line of each test case contains a string of lowercase English letters bb.The third line of each test case contains a string of lowercase English letters cc.It is guaranteed that in each test case these three strings are non-empty and have the same length, which is not exceeding 100100.OutputPrint tt lines with answers for all test cases. For each test case:If it is possible to make string aa equal to string bb print "YES" (without quotes), otherwise print "NO" (without quotes).You can print either lowercase or uppercase letters in the answers.ExampleInputCopy4
aaa
bbb
ccc
abc
bca
bca
aabb
bbaa
baba
imi
mii
iim
OutputCopyNO
YES
YES
NO
NoteIn the first test case; it is impossible to do the swaps so that string aa becomes exactly the same as string bb.In the second test case, you should swap cici with aiai for all possible ii. After the swaps aa becomes "bca", bb becomes "bca" and cc becomes "abc". Here the strings aa and bb are equal.In the third test case, you should swap c1c1 with a1a1, c2c2 with b2b2, c3c3 with b3b3 and c4c4 with a4a4. Then string aa becomes "baba", string bb becomes "baba" and string cc becomes "abab". Here the strings aa and bb are equal.In the fourth test case, it is impossible to do the swaps so that string aa becomes exactly the same as string bb. | [
"implementation",
"strings"
] | a = int(input())
for i in range(a):
q = input()
s = input()
w = input()
d = "YES"
for j in range(len(q)):
if q[j] != w[j] and w[j] != s[j]:
d = "NO"
break
print(d)
| py |
1315 | C | C. Restoring Permutationtime limit per test1 secondmemory limit per test256 megabytesinputstandard inputoutputstandard outputYou are given a sequence b1,b2,…,bnb1,b2,…,bn. Find the lexicographically minimal permutation a1,a2,…,a2na1,a2,…,a2n such that bi=min(a2i−1,a2i)bi=min(a2i−1,a2i), or determine that it is impossible.InputEach test contains one or more test cases. The first line contains the number of test cases tt (1≤t≤1001≤t≤100).The first line of each test case consists of one integer nn — the number of elements in the sequence bb (1≤n≤1001≤n≤100).The second line of each test case consists of nn different integers b1,…,bnb1,…,bn — elements of the sequence bb (1≤bi≤2n1≤bi≤2n).It is guaranteed that the sum of nn by all test cases doesn't exceed 100100.OutputFor each test case, if there is no appropriate permutation, print one number −1−1.Otherwise, print 2n2n integers a1,…,a2na1,…,a2n — required lexicographically minimal permutation of numbers from 11 to 2n2n.ExampleInputCopy5
1
1
2
4 1
3
4 1 3
4
2 3 4 5
5
1 5 7 2 8
OutputCopy1 2
-1
4 5 1 2 3 6
-1
1 3 5 6 7 9 2 4 8 10
| [
"greedy"
] | def solve():
t = int(input())
for i in range(t):
n = int(input())
b = list(map(int, input().split()))
impossible = False
a = [-1] * (2 * n)
numbers = set(b)
for i, x in enumerate(b):
if impossible:
break
a[2 * i] = x
bigger = x + 1
while bigger in numbers:
bigger += 1
if bigger <= 2 * n:
a[2 * i + 1] = bigger
numbers.add(bigger)
else:
impossible = True
if impossible:
print("-1")
else:
print(" ".join(map(str, a)))
solve()
| py |
1296 | D | D. Fight with Monsterstime limit per test1 secondmemory limit per test256 megabytesinputstandard inputoutputstandard outputThere are nn monsters standing in a row numbered from 11 to nn. The ii-th monster has hihi health points (hp). You have your attack power equal to aa hp and your opponent has his attack power equal to bb hp.You and your opponent are fighting these monsters. Firstly, you and your opponent go to the first monster and fight it till his death, then you and your opponent go the second monster and fight it till his death, and so on. A monster is considered dead if its hp is less than or equal to 00.The fight with a monster happens in turns. You hit the monster by aa hp. If it is dead after your hit, you gain one point and you both proceed to the next monster. Your opponent hits the monster by bb hp. If it is dead after his hit, nobody gains a point and you both proceed to the next monster. You have some secret technique to force your opponent to skip his turn. You can use this technique at most kk times in total (for example, if there are two monsters and k=4k=4, then you can use the technique 22 times on the first monster and 11 time on the second monster, but not 22 times on the first monster and 33 times on the second monster).Your task is to determine the maximum number of points you can gain if you use the secret technique optimally.InputThe first line of the input contains four integers n,a,bn,a,b and kk (1≤n≤2⋅105,1≤a,b,k≤1091≤n≤2⋅105,1≤a,b,k≤109) — the number of monsters, your attack power, the opponent's attack power and the number of times you can use the secret technique.The second line of the input contains nn integers h1,h2,…,hnh1,h2,…,hn (1≤hi≤1091≤hi≤109), where hihi is the health points of the ii-th monster.OutputPrint one integer — the maximum number of points you can gain if you use the secret technique optimally.ExamplesInputCopy6 2 3 3
7 10 50 12 1 8
OutputCopy5
InputCopy1 1 100 99
100
OutputCopy1
InputCopy7 4 2 1
1 3 5 4 2 7 6
OutputCopy6
| [
"greedy",
"sortings"
] | import sys, io, os
import math
import bisect
from sys import stdin,stdout
from math import gcd,floor,sqrt,log
from collections import defaultdict as dd
from bisect import bisect_left as bl,bisect_right as br
#input = io.BytesIO(os.read(0,os.fstat(0).st_size)).readline
inp =lambda: int(input())
strng =lambda: input().strip()
jn =lambda x,l: x.join(map(str,l))
strl =lambda: list(input().strip())
mul =lambda: map(int,input().strip().split())
mulf =lambda: map(float,input().strip().split())
seq =lambda: list(map(int,input().strip().split()))
ceil =lambda x: int(x) if(x==int(x)) else int(x)+1
ceildiv=lambda x,d: x//d if(x%d==0) else x//d+1
flush =lambda: stdout.flush()
stdstr =lambda: stdin.readline()
stdint =lambda: int(stdin.readline())
stdpr =lambda x: stdout.write(str(x) + "\n")
printls = lambda l: print(*l, sep=" ")
mod=1000000007
"""
Paste solution below
"""
def solution(h,a,b,k):
for i in range(len(h)):
if h[i]%(a+b) != 0:
h[i] = h[i]%(a+b)
else:
h[i] = a+b
h.sort()
points = 0
for i in range(len(h)):
if h[i] <= a:
points += 1
elif ceildiv(h[i], a) - 1 <= k:
k = k - (ceildiv(h[i], a) - 1)
points += 1
else:
return points
"""while h[i]:
if h[i] <= a:
points += 1
break
elif k:
k -= 1
h[i] -= a
else:
return points
"""
return points
t = 1#inp()
for i in range(t):
n,a,b,k = mul()
h = seq()
ans = solution(h,a,b,k)
print(ans) | py |
1304 | E | E. 1-Trees and Queriestime limit per test4 secondsmemory limit per test512 megabytesinputstandard inputoutputstandard outputGildong was hiking a mountain; walking by millions of trees. Inspired by them, he suddenly came up with an interesting idea for trees in data structures: What if we add another edge in a tree?Then he found that such tree-like graphs are called 1-trees. Since Gildong was bored of solving too many tree problems, he wanted to see if similar techniques in trees can be used in 1-trees as well. Instead of solving it by himself, he's going to test you by providing queries on 1-trees.First, he'll provide you a tree (not 1-tree) with nn vertices, then he will ask you qq queries. Each query contains 55 integers: xx, yy, aa, bb, and kk. This means you're asked to determine if there exists a path from vertex aa to bb that contains exactly kk edges after adding a bidirectional edge between vertices xx and yy. A path can contain the same vertices and same edges multiple times. All queries are independent of each other; i.e. the added edge in a query is removed in the next query.InputThe first line contains an integer nn (3≤n≤1053≤n≤105), the number of vertices of the tree.Next n−1n−1 lines contain two integers uu and vv (1≤u,v≤n1≤u,v≤n, u≠vu≠v) each, which means there is an edge between vertex uu and vv. All edges are bidirectional and distinct.Next line contains an integer qq (1≤q≤1051≤q≤105), the number of queries Gildong wants to ask.Next qq lines contain five integers xx, yy, aa, bb, and kk each (1≤x,y,a,b≤n1≤x,y,a,b≤n, x≠yx≠y, 1≤k≤1091≤k≤109) – the integers explained in the description. It is guaranteed that the edge between xx and yy does not exist in the original tree.OutputFor each query, print "YES" if there exists a path that contains exactly kk edges from vertex aa to bb after adding an edge between vertices xx and yy. Otherwise, print "NO".You can print each letter in any case (upper or lower).ExampleInputCopy5
1 2
2 3
3 4
4 5
5
1 3 1 2 2
1 4 1 3 2
1 4 1 3 3
4 2 3 3 9
5 2 3 3 9
OutputCopyYES
YES
NO
YES
NO
NoteThe image below describes the tree (circles and solid lines) and the added edges for each query (dotted lines). Possible paths for the queries with "YES" answers are: 11-st query: 11 – 33 – 22 22-nd query: 11 – 22 – 33 44-th query: 33 – 44 – 22 – 33 – 44 – 22 – 33 – 44 – 22 – 33 | [
"data structures",
"dfs and similar",
"shortest paths",
"trees"
] | import sys
# Testing out some really weird behaviours in pypy
class RangeQuery:
def __init__(self, data, func=min):
self.func = func
self._data = _data = [list(data)]
i, n = 1, len(_data[0])
while 2 * i <= n:
prev = _data[-1]
_data.append([func(prev[j], prev[j + i]) for j in range(n - 2 * i + 1)])
i <<= 1
def query(self, begin, end):
depth = (end - begin).bit_length() - 1
return self.func(self._data[depth][begin], self._data[depth][end - (1 << depth)])
class LCA:
def __init__(self, root, graph):
self.time = [-1] * len(graph)
self.path = [-1] * len(graph)
P = [-1] * len(graph)
t = -1
dfs = [root]
while dfs:
node = dfs.pop()
self.path[t] = P[node]
self.time[node] = t = t + 1
for nei in graph[node]:
if self.time[nei] == -1:
P[nei] = node
dfs.append(nei)
self.rmq = RangeQuery(self.time[node] for node in self.path)
def __call__(self, a, b):
for _ in range(1): pass
if a == b:
return a
a = self.time[a]
b = self.time[b]
diff = ((b - a) >> 31) & (b - a)
a += diff
b -= diff
return self.path[self.rmq.query(a, b)]
inp = [int(x) for x in sys.stdin.read().split()]; ii = 0
n = inp[ii]; ii += 1
coupl = [[] for _ in range(n)]
for _ in range(n - 1):
u = inp[ii] - 1; ii += 1
v = inp[ii] - 1; ii += 1
coupl[u].append(v)
coupl[v].append(u)
root = 0
lca = LCA(root, coupl)
depth = [-1]*n
depth[root] = 0
bfs = [root]
for node in bfs:
for nei in coupl[node]:
if depth[nei] == -1:
depth[nei] = depth[node] + 1
bfs.append(nei)
def dist(a,b):
for _ in range(0): pass
c = lca(a,b)
return depth[a] + depth[b] - 2 * depth[c]
q = inp[ii]; ii += 1
out = []
for _ in range(q):
x = inp[ii] - 1; ii += 1
y = inp[ii] - 1; ii += 1
a = inp[ii] - 1; ii += 1
b = inp[ii] - 1; ii += 1
k = inp[ii]; ii += 1
dists = [dist(a,b), dist(a,x) + dist(y,b) + 1, dist(a,y) + dist(x,b) + 1]
#dist(a,b) + dist(b,x) # this line makes everything 2 s faster for no reason
for d in dists:
if d - k & 1 == 0 and d <= k:
out.append('YES')
break
else:
out.append('NO')
print('\n'.join(out)) | py |
1288 | D | D. Minimax Problemtime limit per test5 secondsmemory limit per test512 megabytesinputstandard inputoutputstandard outputYou are given nn arrays a1a1, a2a2, ..., anan; each array consists of exactly mm integers. We denote the yy-th element of the xx-th array as ax,yax,y.You have to choose two arrays aiai and ajaj (1≤i,j≤n1≤i,j≤n, it is possible that i=ji=j). After that, you will obtain a new array bb consisting of mm integers, such that for every k∈[1,m]k∈[1,m] bk=max(ai,k,aj,k)bk=max(ai,k,aj,k).Your goal is to choose ii and jj so that the value of mink=1mbkmink=1mbk is maximum possible.InputThe first line contains two integers nn and mm (1≤n≤3⋅1051≤n≤3⋅105, 1≤m≤81≤m≤8) — the number of arrays and the number of elements in each array, respectively.Then nn lines follow, the xx-th line contains the array axax represented by mm integers ax,1ax,1, ax,2ax,2, ..., ax,max,m (0≤ax,y≤1090≤ax,y≤109).OutputPrint two integers ii and jj (1≤i,j≤n1≤i,j≤n, it is possible that i=ji=j) — the indices of the two arrays you have to choose so that the value of mink=1mbkmink=1mbk is maximum possible. If there are multiple answers, print any of them.ExampleInputCopy6 5
5 0 3 1 2
1 8 9 1 3
1 2 3 4 5
9 1 0 3 7
2 3 0 6 3
6 4 1 7 0
OutputCopy1 5
| [
"binary search",
"bitmasks",
"dp"
] | import sys
import io, os
input = io.BytesIO(os.read(0,os.fstat(0).st_size)).readline
from collections import defaultdict
n, m = map(int, input().split())
A = [list(map(int, input().split())) for i in range(n)]
pair = []
for i in range((1<<m)-1):
for j in range(i+1, 1<<m):
if (i|j) == (1<<m)-1:
pair.append((i, j))
def is_ok(x):
C = defaultdict(lambda:-1)
for i in range(n):
k = 0
for j in range(m):
if A[i][j] >= x:
k |= 1<<j
C[k] = i
if C[(1<<m)-1] != -1:
return (C[(1<<m)-1],C[(1<<m)-1])
for k, l in pair:
if C[k] != -1 and C[l] != -1:
return (C[k], C[l])
return (-1, -1)
ok = 0
ng = 10**9+50
while ok+1 < ng:
mid = (ok+ng)//2
if is_ok(mid) != (-1, -1):
ok = mid
else:
ng = mid
i, j = is_ok(ok)
print(i+1, j+1)
| py |
1292 | A | A. NEKO's Maze Gametime limit per test1.5 secondsmemory limit per test256 megabytesinputstandard inputoutputstandard output3R2 as DJ Mashiro - Happiness Breeze Ice - DJ Mashiro is dead or aliveNEKO#ΦωΦ has just got a new maze game on her PC!The game's main puzzle is a maze; in the forms of a 2×n2×n rectangle grid. NEKO's task is to lead a Nekomimi girl from cell (1,1)(1,1) to the gate at (2,n)(2,n) and escape the maze. The girl can only move between cells sharing a common side.However, at some moments during the game, some cells may change their state: either from normal ground to lava (which forbids movement into that cell), or vice versa (which makes that cell passable again). Initially all cells are of the ground type.After hours of streaming, NEKO finally figured out there are only qq such moments: the ii-th moment toggles the state of cell (ri,ci)(ri,ci) (either from ground to lava or vice versa).Knowing this, NEKO wonders, after each of the qq moments, whether it is still possible to move from cell (1,1)(1,1) to cell (2,n)(2,n) without going through any lava cells.Although NEKO is a great streamer and gamer, she still can't get through quizzes and problems requiring large amount of Brain Power. Can you help her?InputThe first line contains integers nn, qq (2≤n≤1052≤n≤105, 1≤q≤1051≤q≤105).The ii-th of qq following lines contains two integers riri, cici (1≤ri≤21≤ri≤2, 1≤ci≤n1≤ci≤n), denoting the coordinates of the cell to be flipped at the ii-th moment.It is guaranteed that cells (1,1)(1,1) and (2,n)(2,n) never appear in the query list.OutputFor each moment, if it is possible to travel from cell (1,1)(1,1) to cell (2,n)(2,n), print "Yes", otherwise print "No". There should be exactly qq answers, one after every update.You can print the words in any case (either lowercase, uppercase or mixed).ExampleInputCopy5 5
2 3
1 4
2 4
2 3
1 4
OutputCopyYes
No
No
No
Yes
NoteWe'll crack down the example test here: After the first query; the girl still able to reach the goal. One of the shortest path ways should be: (1,1)→(1,2)→(1,3)→(1,4)→(1,5)→(2,5)(1,1)→(1,2)→(1,3)→(1,4)→(1,5)→(2,5). After the second query, it's impossible to move to the goal, since the farthest cell she could reach is (1,3)(1,3). After the fourth query, the (2,3)(2,3) is not blocked, but now all the 44-th column is blocked, so she still can't reach the goal. After the fifth query, the column barrier has been lifted, thus she can go to the final goal again. | [
"data structures",
"dsu",
"implementation"
] | n, q = map(int, input().split())
m = [[0 for _ in range(n)], [0 for _ in range(n)]]
blocked = 0
for _ in range(q):
r, c = map(int, input().split())
r = r - 1
c = c - 1
row_to_check = (r + 1) % 2
if c == 0:
columns_to_check = [c, c + 1]
elif c == n - 1:
columns_to_check = [c - 1, c]
else:
columns_to_check = [c - 1, c, c + 1]
m[r][c] = (m[r][c] + 1) % 2
for column_to_check in columns_to_check:
if m[row_to_check][column_to_check] == 1:
if m[r][c]:
blocked += 1
else:
blocked -= 1
print('Yes' if blocked <= 0 else 'No')
| py |
1141 | F1 | F1. Same Sum Blocks (Easy)time limit per test2 secondsmemory limit per test256 megabytesinputstandard inputoutputstandard outputThis problem is given in two editions; which differ exclusively in the constraints on the number nn.You are given an array of integers a[1],a[2],…,a[n].a[1],a[2],…,a[n]. A block is a sequence of contiguous (consecutive) elements a[l],a[l+1],…,a[r]a[l],a[l+1],…,a[r] (1≤l≤r≤n1≤l≤r≤n). Thus, a block is defined by a pair of indices (l,r)(l,r).Find a set of blocks (l1,r1),(l2,r2),…,(lk,rk)(l1,r1),(l2,r2),…,(lk,rk) such that: They do not intersect (i.e. they are disjoint). Formally, for each pair of blocks (li,ri)(li,ri) and (lj,rj(lj,rj) where i≠ji≠j either ri<ljri<lj or rj<lirj<li. For each block the sum of its elements is the same. Formally, a[l1]+a[l1+1]+⋯+a[r1]=a[l2]+a[l2+1]+⋯+a[r2]=a[l1]+a[l1+1]+⋯+a[r1]=a[l2]+a[l2+1]+⋯+a[r2]= ⋯=⋯= a[lk]+a[lk+1]+⋯+a[rk].a[lk]+a[lk+1]+⋯+a[rk]. The number of the blocks in the set is maximum. Formally, there does not exist a set of blocks (l′1,r′1),(l′2,r′2),…,(l′k′,r′k′)(l1′,r1′),(l2′,r2′),…,(lk′′,rk′′) satisfying the above two requirements with k′>kk′>k. The picture corresponds to the first example. Blue boxes illustrate blocks. Write a program to find such a set of blocks.InputThe first line contains integer nn (1≤n≤501≤n≤50) — the length of the given array. The second line contains the sequence of elements a[1],a[2],…,a[n]a[1],a[2],…,a[n] (−105≤ai≤105−105≤ai≤105).OutputIn the first line print the integer kk (1≤k≤n1≤k≤n). The following kk lines should contain blocks, one per line. In each line print a pair of indices li,rili,ri (1≤li≤ri≤n1≤li≤ri≤n) — the bounds of the ii-th block. You can print blocks in any order. If there are multiple answers, print any of them.ExamplesInputCopy7
4 1 2 2 1 5 3
OutputCopy3
7 7
2 3
4 5
InputCopy11
-5 -4 -3 -2 -1 0 1 2 3 4 5
OutputCopy2
3 4
1 1
InputCopy4
1 1 1 1
OutputCopy4
4 4
1 1
2 2
3 3
| [
"greedy"
] | import sys
import threading
input=sys.stdin.readline
from collections import Counter,defaultdict,deque
from heapq import heappush,heappop,heapify
#threading.stack_size(10**8)
#sys.setrecursionlimit(10**6)
def ri():return int(input())
def rs():return input()
def rl():return list(map(int,input().split()))
def rls():return list(input().split())
def main():
n=ri()
a=rl()
d=defaultdict(list)
for l in range(n):
cs=0
for r in range(l,n):
cs+=a[r]
d[cs].append((l,r))
res=[]
for s in d:
d[s].sort(key=lambda x:x[1])
curr=[]
prev=-1
for r in d[s]:
if r[0]>prev:
prev=r[1]
curr.append(r)
if len(curr)>len(res):
res=curr
print(len(res))
for l,r in res:
print(l+1,r+1)
pass
main()
#threading.Thread(target=main).start()
| py |
1141 | G | G. Privatization of Roads in Treelandtime limit per test2 secondsmemory limit per test256 megabytesinputstandard inputoutputstandard outputTreeland consists of nn cities and n−1n−1 roads. Each road is bidirectional and connects two distinct cities. From any city you can get to any other city by roads. Yes, you are right — the country's topology is an undirected tree.There are some private road companies in Treeland. The government decided to sell roads to the companies. Each road will belong to one company and a company can own multiple roads.The government is afraid to look unfair. They think that people in a city can consider them unfair if there is one company which owns two or more roads entering the city. The government wants to make such privatization that the number of such cities doesn't exceed kk and the number of companies taking part in the privatization is minimal.Choose the number of companies rr such that it is possible to assign each road to one company in such a way that the number of cities that have two or more roads of one company is at most kk. In other words, if for a city all the roads belong to the different companies then the city is good. Your task is to find the minimal rr that there is such assignment to companies from 11 to rr that the number of cities which are not good doesn't exceed kk. The picture illustrates the first example (n=6,k=2n=6,k=2). The answer contains r=2r=2 companies. Numbers on the edges denote edge indices. Edge colors mean companies: red corresponds to the first company, blue corresponds to the second company. The gray vertex (number 33) is not good. The number of such vertices (just one) doesn't exceed k=2k=2. It is impossible to have at most k=2k=2 not good cities in case of one company. InputThe first line contains two integers nn and kk (2≤n≤200000,0≤k≤n−12≤n≤200000,0≤k≤n−1) — the number of cities and the maximal number of cities which can have two or more roads belonging to one company.The following n−1n−1 lines contain roads, one road per line. Each line contains a pair of integers xixi, yiyi (1≤xi,yi≤n1≤xi,yi≤n), where xixi, yiyi are cities connected with the ii-th road.OutputIn the first line print the required rr (1≤r≤n−11≤r≤n−1). In the second line print n−1n−1 numbers c1,c2,…,cn−1c1,c2,…,cn−1 (1≤ci≤r1≤ci≤r), where cici is the company to own the ii-th road. If there are multiple answers, print any of them.ExamplesInputCopy6 2
1 4
4 3
3 5
3 6
5 2
OutputCopy2
1 2 1 1 2 InputCopy4 2
3 1
1 4
1 2
OutputCopy1
1 1 1 InputCopy10 2
10 3
1 2
1 3
1 4
2 5
2 6
2 7
3 8
3 9
OutputCopy3
1 1 2 3 2 3 1 3 1 | [
"binary search",
"constructive algorithms",
"dfs and similar",
"graphs",
"greedy",
"trees"
] | import sys
input=sys.stdin.buffer.readline
from collections import deque
def dfs():
visited=[0 for i in range(n+1)]
queue=deque()
queue.append([1,0])
add=0
while(queue):
par=queue.popleft()
if(visited[par[0]]==1):
continue
visited[par[0]]=1
add=(marked[par[1]]+1)%fix
for child in graph[par[0]]:
if(visited[child[0]]==0):
marked[child[1]]=add
add=(add+1)%fix
queue.append([child[0],child[1]])
n,k=map(int,input().split())
graph=[[] for i in range(n+1)]
counted=[0 for i in range(n+1)]
for i in range(n-1):
a,b=map(int,input().split())
counted[a]+=1
counted[b]+=1
graph[a].append([b,i])
graph[b].append([a,i])
arr=[]
for i in range(1,n+1):
arr.append(counted[i])
arr.sort(reverse=True)
fix=arr[k]
marked=[0 for i in range(n+1)]
marked[0]=-1
dfs()
print(max(marked)+1)
for i in range(n-1):
print(marked[i]+1,end=' ')
print() | py |
1323 | B | B. Count Subrectanglestime limit per test1 secondmemory limit per test512 megabytesinputstandard inputoutputstandard outputYou are given an array aa of length nn and array bb of length mm both consisting of only integers 00 and 11. Consider a matrix cc of size n×mn×m formed by following rule: ci,j=ai⋅bjci,j=ai⋅bj (i.e. aiai multiplied by bjbj). It's easy to see that cc consists of only zeroes and ones too.How many subrectangles of size (area) kk consisting only of ones are there in cc?A subrectangle is an intersection of a consecutive (subsequent) segment of rows and a consecutive (subsequent) segment of columns. I.e. consider four integers x1,x2,y1,y2x1,x2,y1,y2 (1≤x1≤x2≤n1≤x1≤x2≤n, 1≤y1≤y2≤m1≤y1≤y2≤m) a subrectangle c[x1…x2][y1…y2]c[x1…x2][y1…y2] is an intersection of the rows x1,x1+1,x1+2,…,x2x1,x1+1,x1+2,…,x2 and the columns y1,y1+1,y1+2,…,y2y1,y1+1,y1+2,…,y2.The size (area) of a subrectangle is the total number of cells in it.InputThe first line contains three integers nn, mm and kk (1≤n,m≤40000,1≤k≤n⋅m1≤n,m≤40000,1≤k≤n⋅m), length of array aa, length of array bb and required size of subrectangles.The second line contains nn integers a1,a2,…,ana1,a2,…,an (0≤ai≤10≤ai≤1), elements of aa.The third line contains mm integers b1,b2,…,bmb1,b2,…,bm (0≤bi≤10≤bi≤1), elements of bb.OutputOutput single integer — the number of subrectangles of cc with size (area) kk consisting only of ones.ExamplesInputCopy3 3 2
1 0 1
1 1 1
OutputCopy4
InputCopy3 5 4
1 1 1
1 1 1 1 1
OutputCopy14
NoteIn first example matrix cc is: There are 44 subrectangles of size 22 consisting of only ones in it: In second example matrix cc is: | [
"binary search",
"greedy",
"implementation"
] | # 10:59-
import sys
input = lambda: sys.stdin.readline().strip()
from collections import Counter
N,M,K = map(int, input().split())
A = list(map(int, input().split()))
B = list(map(int, input().split()))
recs = set()
for i in range(1,int(K**0.5)+1):
if K%i==0:
recs.add((i,K//i))
recs.add((K//i,i))
def get_segs(A):
segs = [0]
for a in A:
if a==0:
if segs[-1]>0:
segs.append(0)
else:
segs[-1]+=1
if segs[-1]==0:
segs.pop()
return Counter(segs)
def get_cnt(w,h):
ans=0
for w1,h1 in recs:
if w1>w or h1>h:continue
ans+=(w-w1+1)*(h-h1+1)
return ans
AC,BC=get_segs(A),get_segs(B)
ans=0
for ak,av in AC.items():
for bk,bv in BC.items():
if ak*bk<K:continue
ans+=get_cnt(ak,bk)*av*bv
print(ans)
| py |
1311 | D | D. Three Integerstime limit per test2 secondsmemory limit per test256 megabytesinputstandard inputoutputstandard outputYou are given three integers a≤b≤ca≤b≤c.In one move, you can add +1+1 or −1−1 to any of these integers (i.e. increase or decrease any number by one). You can perform such operation any (possibly, zero) number of times, you can even perform this operation several times with one number. Note that you cannot make non-positive numbers using such operations.You have to perform the minimum number of such operations in order to obtain three integers A≤B≤CA≤B≤C such that BB is divisible by AA and CC is divisible by BB.You have to answer tt independent test cases. InputThe first line of the input contains one integer tt (1≤t≤1001≤t≤100) — the number of test cases.The next tt lines describe test cases. Each test case is given on a separate line as three space-separated integers a,ba,b and cc (1≤a≤b≤c≤1041≤a≤b≤c≤104).OutputFor each test case, print the answer. In the first line print resres — the minimum number of operations you have to perform to obtain three integers A≤B≤CA≤B≤C such that BB is divisible by AA and CC is divisible by BB. On the second line print any suitable triple A,BA,B and CC.ExampleInputCopy8
1 2 3
123 321 456
5 10 15
15 18 21
100 100 101
1 22 29
3 19 38
6 30 46
OutputCopy1
1 1 3
102
114 228 456
4
4 8 16
6
18 18 18
1
100 100 100
7
1 22 22
2
1 19 38
8
6 24 48
| [
"brute force",
"math"
] | import sys
import math
import itertools
input = sys.stdin.readline
def solve():
a , b , c = map(int , input().split())
ansA , ansB , ansC = 1 , 1 , 1
ans = (a-1)+(b-1)+(c-1)
i = 1
while i <= 2*a:
tmpA = i
tmpB = i
tmpC = tmpB * (c//tmpB)
tmpAns = abs(a-tmpA)+abs(b-tmpB)+abs(c-tmpC)
if tmpAns < ans:
ans = tmpAns
ansA,ansB,ansC = tmpA,tmpB,tmpC
tmpC = tmpB * (c//tmpB) + tmpB
tmpAns = abs(a-tmpA)+abs(b-tmpB)+abs(c-tmpC)
if tmpAns < ans:
ans = tmpAns
ansA,ansB,ansC = tmpA,tmpB,tmpC
while tmpB <= 2*b:
tmpB += i
tmpC = tmpB * (c//tmpB)
tmpAns = abs(a-tmpA)+abs(b-tmpB)+abs(c-tmpC)
if tmpAns < ans:
ans = tmpAns
ansA,ansB,ansC = tmpA,tmpB,tmpC
tmpC = tmpB * (c//tmpB) + tmpB
tmpAns = abs(a-tmpA)+abs(b-tmpB)+abs(c-tmpC)
if tmpAns < ans:
ans = tmpAns
ansA,ansB,ansC = tmpA,tmpB,tmpC
i += 1
print(ans)
print(ansA,ansB,ansC)
for _ in range(int(input())):
solve()
| py |
1307 | C | C. Cow and Messagetime limit per test2 secondsmemory limit per test256 megabytesinputstandard inputoutputstandard outputBessie the cow has just intercepted a text that Farmer John sent to Burger Queen! However; Bessie is sure that there is a secret message hidden inside.The text is a string ss of lowercase Latin letters. She considers a string tt as hidden in string ss if tt exists as a subsequence of ss whose indices form an arithmetic progression. For example, the string aab is hidden in string aaabb because it occurs at indices 11, 33, and 55, which form an arithmetic progression with a common difference of 22. Bessie thinks that any hidden string that occurs the most times is the secret message. Two occurrences of a subsequence of SS are distinct if the sets of indices are different. Help her find the number of occurrences of the secret message!For example, in the string aaabb, a is hidden 33 times, b is hidden 22 times, ab is hidden 66 times, aa is hidden 33 times, bb is hidden 11 time, aab is hidden 22 times, aaa is hidden 11 time, abb is hidden 11 time, aaab is hidden 11 time, aabb is hidden 11 time, and aaabb is hidden 11 time. The number of occurrences of the secret message is 66.InputThe first line contains a string ss of lowercase Latin letters (1≤|s|≤1051≤|s|≤105) — the text that Bessie intercepted.OutputOutput a single integer — the number of occurrences of the secret message.ExamplesInputCopyaaabb
OutputCopy6
InputCopyusaco
OutputCopy1
InputCopylol
OutputCopy2
NoteIn the first example; these are all the hidden strings and their indice sets: a occurs at (1)(1), (2)(2), (3)(3) b occurs at (4)(4), (5)(5) ab occurs at (1,4)(1,4), (1,5)(1,5), (2,4)(2,4), (2,5)(2,5), (3,4)(3,4), (3,5)(3,5) aa occurs at (1,2)(1,2), (1,3)(1,3), (2,3)(2,3) bb occurs at (4,5)(4,5) aab occurs at (1,3,5)(1,3,5), (2,3,4)(2,3,4) aaa occurs at (1,2,3)(1,2,3) abb occurs at (3,4,5)(3,4,5) aaab occurs at (1,2,3,4)(1,2,3,4) aabb occurs at (2,3,4,5)(2,3,4,5) aaabb occurs at (1,2,3,4,5)(1,2,3,4,5) Note that all the sets of indices are arithmetic progressions.In the second example, no hidden string occurs more than once.In the third example, the hidden string is the letter l. | [
"brute force",
"dp",
"math",
"strings"
] | d={}
s=set()
for x in input():
for y in s:d[x,y]=d.get((x,y),0)+d[y]
s|={x};d[x]=d.get(x,0)+1
print(max(d.values()))
num_inp=lambda: int(input())
arr_inp=lambda: list(map(int,input().split()))
sp_inp=lambda: map(int,input().split())
str_inp=lambda:input()
| py |
1288 | E | E. Messenger Simulatortime limit per test3 secondsmemory limit per test256 megabytesinputstandard inputoutputstandard outputPolycarp is a frequent user of the very popular messenger. He's chatting with his friends all the time. He has nn friends, numbered from 11 to nn.Recall that a permutation of size nn is an array of size nn such that each integer from 11 to nn occurs exactly once in this array.So his recent chat list can be represented with a permutation pp of size nn. p1p1 is the most recent friend Polycarp talked to, p2p2 is the second most recent and so on.Initially, Polycarp's recent chat list pp looks like 1,2,…,n1,2,…,n (in other words, it is an identity permutation).After that he receives mm messages, the jj-th message comes from the friend ajaj. And that causes friend ajaj to move to the first position in a permutation, shifting everyone between the first position and the current position of ajaj by 11. Note that if the friend ajaj is in the first position already then nothing happens.For example, let the recent chat list be p=[4,1,5,3,2]p=[4,1,5,3,2]: if he gets messaged by friend 33, then pp becomes [3,4,1,5,2][3,4,1,5,2]; if he gets messaged by friend 44, then pp doesn't change [4,1,5,3,2][4,1,5,3,2]; if he gets messaged by friend 22, then pp becomes [2,4,1,5,3][2,4,1,5,3]. For each friend consider all position he has been at in the beginning and after receiving each message. Polycarp wants to know what were the minimum and the maximum positions.InputThe first line contains two integers nn and mm (1≤n,m≤3⋅1051≤n,m≤3⋅105) — the number of Polycarp's friends and the number of received messages, respectively.The second line contains mm integers a1,a2,…,ama1,a2,…,am (1≤ai≤n1≤ai≤n) — the descriptions of the received messages.OutputPrint nn pairs of integers. For each friend output the minimum and the maximum positions he has been in the beginning and after receiving each message.ExamplesInputCopy5 4
3 5 1 4
OutputCopy1 3
2 5
1 4
1 5
1 5
InputCopy4 3
1 2 4
OutputCopy1 3
1 2
3 4
1 4
NoteIn the first example; Polycarp's recent chat list looks like this: [1,2,3,4,5][1,2,3,4,5] [3,1,2,4,5][3,1,2,4,5] [5,3,1,2,4][5,3,1,2,4] [1,5,3,2,4][1,5,3,2,4] [4,1,5,3,2][4,1,5,3,2] So, for example, the positions of the friend 22 are 2,3,4,4,52,3,4,4,5, respectively. Out of these 22 is the minimum one and 55 is the maximum one. Thus, the answer for the friend 22 is a pair (2,5)(2,5).In the second example, Polycarp's recent chat list looks like this: [1,2,3,4][1,2,3,4] [1,2,3,4][1,2,3,4] [2,1,3,4][2,1,3,4] [4,2,1,3][4,2,1,3] | [
"data structures"
] | import sys
class segmenttree:
def __init__(self, n, default=0, func=lambda a, b: a + b):
self.tree, self.n, self.func, self.default = [0] * (2 * n), n, func, default
def fill(self, arr):
self.tree[self.n:] = arr
for i in range(self.n - 1, 0, -1):
self.tree[i] = self.func(self.tree[i << 1], self.tree[(i << 1) + 1])
# get interval[l,r)
def query(self, l, r):
res = self.default
l += self.n
r += self.n
while l < r:
if l & 1:
res = self.func(res, self.tree[l])
l += 1
if r & 1:
r -= 1
res = self.func(res, self.tree[r])
l >>= 1
r >>= 1
return res
def __setitem__(self, ix, val):
ix += self.n
self.tree[ix] = val
while ix > 1:
self.tree[ix >> 1] = self.func(self.tree[ix], self.tree[ix ^ 1])
ix >>= 1
def __getitem__(self, item):
return self.tree[item + self.n]
input = lambda: sys.stdin.buffer.readline().decode().strip()
n, m = map(int, input().split())
a, mi, ma, lst = [int(x) - 1 for x in input().split()], list(range(1, n + 1)), list(range(1, n + 1)), [-1] * n
tree, tree2 = segmenttree(m), segmenttree(n)
for i in range(m):
mi[a[i]] = 1
if lst[a[i]] != -1:
ma[a[i]] = max(ma[a[i]], tree.query(lst[a[i]], i))
tree[lst[a[i]]] = 0
else:
ma[a[i]] += tree2.query(a[i] + 1, n)
tree2[a[i]] = tree[i] = 1
lst[a[i]] = i
for i in range(n):
if lst[i] == -1:
ma[i] += tree2.query(i + 1, n)
else:
ma[i] = max(ma[i], tree.query(lst[i], m))
print(*[f'{mi[i]} {ma[i]}' for i in range(n)], sep='\n')
| py |
1316 | C | C. Primitive Primestime limit per test1.5 secondsmemory limit per test256 megabytesinputstandard inputoutputstandard outputIt is Professor R's last class of his teaching career. Every time Professor R taught a class; he gave a special problem for the students to solve. You being his favourite student, put your heart into solving it one last time.You are given two polynomials f(x)=a0+a1x+⋯+an−1xn−1f(x)=a0+a1x+⋯+an−1xn−1 and g(x)=b0+b1x+⋯+bm−1xm−1g(x)=b0+b1x+⋯+bm−1xm−1, with positive integral coefficients. It is guaranteed that the cumulative GCD of the coefficients is equal to 11 for both the given polynomials. In other words, gcd(a0,a1,…,an−1)=gcd(b0,b1,…,bm−1)=1gcd(a0,a1,…,an−1)=gcd(b0,b1,…,bm−1)=1. Let h(x)=f(x)⋅g(x)h(x)=f(x)⋅g(x). Suppose that h(x)=c0+c1x+⋯+cn+m−2xn+m−2h(x)=c0+c1x+⋯+cn+m−2xn+m−2. You are also given a prime number pp. Professor R challenges you to find any tt such that ctct isn't divisible by pp. He guarantees you that under these conditions such tt always exists. If there are several such tt, output any of them.As the input is quite large, please use fast input reading methods.InputThe first line of the input contains three integers, nn, mm and pp (1≤n,m≤106,2≤p≤1091≤n,m≤106,2≤p≤109), — nn and mm are the number of terms in f(x)f(x) and g(x)g(x) respectively (one more than the degrees of the respective polynomials) and pp is the given prime number.It is guaranteed that pp is prime.The second line contains nn integers a0,a1,…,an−1a0,a1,…,an−1 (1≤ai≤1091≤ai≤109) — aiai is the coefficient of xixi in f(x)f(x).The third line contains mm integers b0,b1,…,bm−1b0,b1,…,bm−1 (1≤bi≤1091≤bi≤109) — bibi is the coefficient of xixi in g(x)g(x).OutputPrint a single integer tt (0≤t≤n+m−20≤t≤n+m−2) — the appropriate power of xx in h(x)h(x) whose coefficient isn't divisible by the given prime pp. If there are multiple powers of xx that satisfy the condition, print any.ExamplesInputCopy3 2 2
1 1 2
2 1
OutputCopy1
InputCopy2 2 999999937
2 1
3 1
OutputCopy2NoteIn the first test case; f(x)f(x) is 2x2+x+12x2+x+1 and g(x)g(x) is x+2x+2, their product h(x)h(x) being 2x3+5x2+3x+22x3+5x2+3x+2, so the answer can be 1 or 2 as both 3 and 5 aren't divisible by 2.In the second test case, f(x)f(x) is x+2x+2 and g(x)g(x) is x+3x+3, their product h(x)h(x) being x2+5x+6x2+5x+6, so the answer can be any of the powers as no coefficient is divisible by the given prime. | [
"constructive algorithms",
"math",
"ternary search"
] | import os
import sys
from io import BytesIO, IOBase
from collections import Counter
import math as mt
BUFSIZE = 8192
class FastIO(IOBase):
newlines = 0
def __init__(self, file):
self._fd = file.fileno()
self.buffer = BytesIO()
self.writable = "x" in file.mode or "r" not in file.mode
self.write = self.buffer.write if self.writable else None
def read(self):
while True:
b = os.read(self._fd, max(os.fstat(self._fd).st_size, BUFSIZE))
if not b:
break
ptr = self.buffer.tell()
self.buffer.seek(0, 2), self.buffer.write(b), self.buffer.seek(ptr)
self.newlines = 0
return self.buffer.read()
def readline(self):
while self.newlines == 0:
b = os.read(self._fd, max(os.fstat(self._fd).st_size, BUFSIZE))
self.newlines = b.count(b"\n") + (not b)
ptr = self.buffer.tell()
self.buffer.seek(0, 2), self.buffer.write(b), self.buffer.seek(ptr)
self.newlines -= 1
return self.buffer.readline()
def flush(self):
if self.writable:
os.write(self._fd, self.buffer.getvalue())
self.buffer.truncate(0), self.buffer.seek(0)
class IOWrapper(IOBase):
def __init__(self, file):
self.buffer = FastIO(file)
self.flush = self.buffer.flush
self.writable = self.buffer.writable
self.write = lambda s: self.buffer.write(s.encode("ascii"))
self.read = lambda: self.buffer.read().decode("ascii")
self.readline = lambda: self.buffer.readline().decode("ascii")
sys.stdin, sys.stdout = IOWrapper(sys.stdin), IOWrapper(sys.stdout)
def gcd(a, b):
if a == 0:
return b
return gcd(b % a, a)
def lcm(a, b):
return (a * b) / gcd(a, b)
mod = int(1e9) + 7
def power(k, n):
if n == 0:
return 1
if n % 2:
return (power(k, n - 1) * k) % mod
t = power(k, n // 2)
return (t * t) % mod
def totalPrimeFactors(n):
count = 0
if (n % 2) == 0:
count += 1
while (n % 2) == 0:
n //= 2
i = 3
while i * i <= n:
if (n % i) == 0:
count += 1
while (n % i) == 0:
n //= i
i += 2
if n > 2:
count += 1
return count
# #MAXN = int(1e7 + 1)
# # spf = [0 for i in range(MAXN)]
#
#
# def sieve():
# spf[1] = 1
# for i in range(2, MAXN):
# spf[i] = i
# for i in range(4, MAXN, 2):
# spf[i] = 2
#
# for i in range(3, mt.ceil(mt.sqrt(MAXN))):
# if (spf[i] == i):
# for j in range(i * i, MAXN, i):
# if (spf[j] == j):
# spf[j] = i
#
#
# def getFactorization(x):
# ret = 0
# while (x != 1):
# k = spf[x]
# ret += 1
# # ret.add(spf[x])
# while x % k == 0:
# x //= k
#
# return ret
# Driver code
# precalculating Smallest Prime Factor
# sieve()
def main():
n,m,p=map(int, input().split())
a=list(map(int, input().split()))
b=list(map(int, input().split()))
f,s=-1, -1
for i in range(n):
if a[i]%p:
f=i
break
for i in range(m):
if b[i]%p:
s=i
break
print(f+ s)
return
if __name__ == "__main__":
main()
| py |
1305 | D | D. Kuroni and the Celebrationtime limit per test1 secondmemory limit per test256 megabytesinputstandard inputoutputstandard outputThis is an interactive problem.After getting AC after 13 Time Limit Exceeded verdicts on a geometry problem; Kuroni went to an Italian restaurant to celebrate this holy achievement. Unfortunately, the excess sauce disoriented him, and he's now lost!The United States of America can be modeled as a tree (why though) with nn vertices. The tree is rooted at vertex rr, wherein lies Kuroni's hotel.Kuroni has a phone app designed to help him in such emergency cases. To use the app, he has to input two vertices uu and vv, and it'll return a vertex ww, which is the lowest common ancestor of those two vertices.However, since the phone's battery has been almost drained out from live-streaming Kuroni's celebration party, he could only use the app at most ⌊n2⌋⌊n2⌋ times. After that, the phone would die and there will be nothing left to help our dear friend! :(As the night is cold and dark, Kuroni needs to get back, so that he can reunite with his comfy bed and pillow(s). Can you help him figure out his hotel's location?InteractionThe interaction starts with reading a single integer nn (2≤n≤10002≤n≤1000), the number of vertices of the tree.Then you will read n−1n−1 lines, the ii-th of them has two integers xixi and yiyi (1≤xi,yi≤n1≤xi,yi≤n, xi≠yixi≠yi), denoting there is an edge connecting vertices xixi and yiyi. It is guaranteed that the edges will form a tree.Then you can make queries of type "? u v" (1≤u,v≤n1≤u,v≤n) to find the lowest common ancestor of vertex uu and vv.After the query, read the result ww as an integer.In case your query is invalid or you asked more than ⌊n2⌋⌊n2⌋ queries, the program will print −1−1 and will finish interaction. You will receive a Wrong answer verdict. Make sure to exit immediately to avoid getting other verdicts.When you find out the vertex rr, print "! rr" and quit after that. This query does not count towards the ⌊n2⌋⌊n2⌋ limit.Note that the tree is fixed beforehand and will not change during the queries, i.e. the interactor is not adaptive.After printing any query do not forget to print end of line and flush the output. Otherwise, you might get Idleness limit exceeded. To do this, use: fflush(stdout) or cout.flush() in C++; System.out.flush() in Java; flush(output) in Pascal; stdout.flush() in Python; see the documentation for other languages.HacksTo hack, use the following format:The first line should contain two integers nn and rr (2≤n≤10002≤n≤1000, 1≤r≤n1≤r≤n), denoting the number of vertices and the vertex with Kuroni's hotel.The ii-th of the next n−1n−1 lines should contain two integers xixi and yiyi (1≤xi,yi≤n1≤xi,yi≤n) — denoting there is an edge connecting vertex xixi and yiyi.The edges presented should form a tree.ExampleInputCopy6
1 4
4 2
5 3
6 3
2 3
3
4
4
OutputCopy
? 5 6
? 3 1
? 1 2
! 4NoteNote that the example interaction contains extra empty lines so that it's easier to read. The real interaction doesn't contain any empty lines and you shouldn't print any extra empty lines as well.The image below demonstrates the tree in the sample test: | [
"constructive algorithms",
"dfs and similar",
"interactive",
"trees"
] | from sys import stdin, stdout, setrecursionlimit
input = stdin.readline
# stdout.flush()
# import string
# characters = string.ascii_lowercase
# digits = string.digits
# setrecursionlimit(int(1e6))
# dir = [-1,0,1,0,-1]
# moves = 'NESW'
from random import shuffle, randint
inf = float('inf')
from functools import cmp_to_key
from collections import defaultdict as dd
from collections import Counter, deque
from heapq import *
import math
from math import floor, ceil, sqrt
def geti(): return map(int, input().strip().split())
def getl(): return list(map(int, input().strip().split()))
def getis(): return map(str, input().strip().split())
def getls(): return list(map(str, input().strip().split()))
def gets(): return input().strip()
def geta(): return int(input())
def print_s(s): stdout.write(s+'\n')
def query(u, v, q):
q[0] += 1
print("? {} {}".format(u, v))
stdout.flush()
now = geta()
if now == -1:
exit()
return now
def solve():
n = geta()
q = [0]
limit = n//2
edges = dd(list)
for i in range(n-1):
u, v = geti()
edges[u].append(v)
edges[v].append(u)
distance = dd(int)
maxi = [-inf]
def dfs(root, dist = 0, par = None):
if maxi[0] == -inf or distance[maxi[0]] < dist:
maxi[0] = root
distance[root] = dist
for node in edges[root]:
if node != par:
dfs(node, dist + 1, root)
leaf = set()
leaves = set()
for i in range(1, n+1):
if len(edges[i]) == 1 and i not in leaf:
maxi[0] = -inf
distance.clear()
dfs(i)
leaf.add(i)
leaf.add(maxi[0])
leaves.add(i)
leaves.add(maxi[0])
leaf = list(leaf)
while len(leaf) != 1 and q[0] < limit:
new = set()
for i in range(1, len(leaf), 2):
if q[0] > limit:
break
now = query(leaf[i], leaf[i-1], q)
if now in leaves:
print("!", now)
stdout.flush()
exit()
new.add(now)
if len(leaf) & 1:
new.add(leaf[-1])
leaf = list(new)
print("!", leaf[0])
stdout.flush()
if __name__=='__main__':
solve()
#
# 5 5
# 1 2
# 1 3
# 4 1
# 4 5
# 5
# |
# 4
# /
# 1
# / \
# 2 3
| py |
1141 | G | G. Privatization of Roads in Treelandtime limit per test2 secondsmemory limit per test256 megabytesinputstandard inputoutputstandard outputTreeland consists of nn cities and n−1n−1 roads. Each road is bidirectional and connects two distinct cities. From any city you can get to any other city by roads. Yes, you are right — the country's topology is an undirected tree.There are some private road companies in Treeland. The government decided to sell roads to the companies. Each road will belong to one company and a company can own multiple roads.The government is afraid to look unfair. They think that people in a city can consider them unfair if there is one company which owns two or more roads entering the city. The government wants to make such privatization that the number of such cities doesn't exceed kk and the number of companies taking part in the privatization is minimal.Choose the number of companies rr such that it is possible to assign each road to one company in such a way that the number of cities that have two or more roads of one company is at most kk. In other words, if for a city all the roads belong to the different companies then the city is good. Your task is to find the minimal rr that there is such assignment to companies from 11 to rr that the number of cities which are not good doesn't exceed kk. The picture illustrates the first example (n=6,k=2n=6,k=2). The answer contains r=2r=2 companies. Numbers on the edges denote edge indices. Edge colors mean companies: red corresponds to the first company, blue corresponds to the second company. The gray vertex (number 33) is not good. The number of such vertices (just one) doesn't exceed k=2k=2. It is impossible to have at most k=2k=2 not good cities in case of one company. InputThe first line contains two integers nn and kk (2≤n≤200000,0≤k≤n−12≤n≤200000,0≤k≤n−1) — the number of cities and the maximal number of cities which can have two or more roads belonging to one company.The following n−1n−1 lines contain roads, one road per line. Each line contains a pair of integers xixi, yiyi (1≤xi,yi≤n1≤xi,yi≤n), where xixi, yiyi are cities connected with the ii-th road.OutputIn the first line print the required rr (1≤r≤n−11≤r≤n−1). In the second line print n−1n−1 numbers c1,c2,…,cn−1c1,c2,…,cn−1 (1≤ci≤r1≤ci≤r), where cici is the company to own the ii-th road. If there are multiple answers, print any of them.ExamplesInputCopy6 2
1 4
4 3
3 5
3 6
5 2
OutputCopy2
1 2 1 1 2 InputCopy4 2
3 1
1 4
1 2
OutputCopy1
1 1 1 InputCopy10 2
10 3
1 2
1 3
1 4
2 5
2 6
2 7
3 8
3 9
OutputCopy3
1 1 2 3 2 3 1 3 1 | [
"binary search",
"constructive algorithms",
"dfs and similar",
"graphs",
"greedy",
"trees"
] | import sys
from collections import deque
def norm_edge(i, j):
return (i, j) if i < j else (j, i)
inpt = sys.stdin.read().split('\n')
n, k = map(int, inpt[0].split())
edges = []
for edge_str in inpt[1:n]:
i, j = tuple(map(lambda x: int(x) - 1, edge_str.split()))
edges.append(norm_edge(i, j))
degree = [0 for _ in range(n)]
neigh = [[] for _ in range(n)]
for i, j in edges:
degree[i] += 1
degree[j] += 1
neigh[i].append(j)
neigh[j].append(i)
r = sorted(degree)[-(k + 1)]
print(r)
visited = set()
color = {}
q = deque([0])
visited = set([0])
while len(q) > 0:
i = q.popleft()
used_color = None
for j in neigh[i]:
if j not in visited:
visited.add(j)
q.append(j)
e = norm_edge(i, j)
if used_color is None and e in color:
used_color = color[e]
next_color = 0
for j in neigh[i]:
e = norm_edge(i, j)
if e in color:
continue
if next_color == used_color:
next_color += 1
color[e] = next_color % r if next_color >= r else next_color
next_color += 1
print(' '.join([str(color[e] + 1) for e in edges]))
# for i in range(n):
# print("Node", i)
# for j in neigh[i]:
# e = norm_edge(i, j)
# print(" ", e, color[e])
| py |
1305 | F | F. Kuroni and the Punishmenttime limit per test2.5 secondsmemory limit per test256 megabytesinputstandard inputoutputstandard outputKuroni is very angry at the other setters for using him as a theme! As a punishment; he forced them to solve the following problem:You have an array aa consisting of nn positive integers. An operation consists of choosing an element and either adding 11 to it or subtracting 11 from it, such that the element remains positive. We say the array is good if the greatest common divisor of all its elements is not 11. Find the minimum number of operations needed to make the array good.Unable to match Kuroni's intellect, the setters failed to solve the problem. Help them escape from Kuroni's punishment!InputThe first line contains an integer nn (2≤n≤2⋅1052≤n≤2⋅105) — the number of elements in the array.The second line contains nn integers a1,a2,…,ana1,a2,…,an. (1≤ai≤10121≤ai≤1012) — the elements of the array.OutputPrint a single integer — the minimum number of operations required to make the array good.ExamplesInputCopy3
6 2 4
OutputCopy0
InputCopy5
9 8 7 3 1
OutputCopy4
NoteIn the first example; the first array is already good; since the greatest common divisor of all the elements is 22.In the second example, we may apply the following operations: Add 11 to the second element, making it equal to 99. Subtract 11 from the third element, making it equal to 66. Add 11 to the fifth element, making it equal to 22. Add 11 to the fifth element again, making it equal to 33. The greatest common divisor of all elements will then be equal to 33, so the array will be good. It can be shown that no sequence of three or less operations can make the array good. | [
"math",
"number theory",
"probabilities"
] | import random
# Sieve
sieve_primes = []
PRIME_LIMIT = int(1e6)
sieve = [False for i in range(0,PRIME_LIMIT)]
for i in range(2,PRIME_LIMIT):
if not sieve[i]:
sieve_primes.append(i)
for j in range(i*i, PRIME_LIMIT, i):
sieve[j]=True
# Input
n = int(input())
arr=list(map(int, input().split()))
# Construct search space Primes
primes = set()
def addPrime(num):
for sieve_prime in sieve_primes:
if num%sieve_prime == 0:
primes.add(sieve_prime)
while num%sieve_prime == 0:
num//=sieve_prime
if num>1:
primes.add(num)
# (Could use probability calculations here to reduce search space)
random.shuffle(arr)
arr_set=list(set(arr))
for num in arr_set[:20]:
if num > 1:
addPrime(num)
addPrime(num+1)
if num > 2:
addPrime(num-1)
# Find answer
ans = n
# Function to find answer based on input prime
def findAns(prime):
global ans
curr_ans = 0
for num in arr:
if num<prime:
curr_ans += prime-num
else:
curr_ans += min(num%prime, prime - num%prime)
if curr_ans >= ans:
return
ans = min(curr_ans, ans)
for prime in primes:
findAns(prime)
if ans == 0: break
print(ans) | py |
1322 | B | B. Presenttime limit per test3 secondsmemory limit per test512 megabytesinputstandard inputoutputstandard outputCatherine received an array of integers as a gift for March 8. Eventually she grew bored with it; and she started calculated various useless characteristics for it. She succeeded to do it for each one she came up with. But when she came up with another one — xor of all pairwise sums of elements in the array, she realized that she couldn't compute it for a very large array, thus she asked for your help. Can you do it? Formally, you need to compute(a1+a2)⊕(a1+a3)⊕…⊕(a1+an)⊕(a2+a3)⊕…⊕(a2+an)…⊕(an−1+an)(a1+a2)⊕(a1+a3)⊕…⊕(a1+an)⊕(a2+a3)⊕…⊕(a2+an)…⊕(an−1+an)Here x⊕yx⊕y is a bitwise XOR operation (i.e. xx ^ yy in many modern programming languages). You can read about it in Wikipedia: https://en.wikipedia.org/wiki/Exclusive_or#Bitwise_operation.InputThe first line contains a single integer nn (2≤n≤4000002≤n≤400000) — the number of integers in the array.The second line contains integers a1,a2,…,ana1,a2,…,an (1≤ai≤1071≤ai≤107).OutputPrint a single integer — xor of all pairwise sums of integers in the given array.ExamplesInputCopy2
1 2
OutputCopy3InputCopy3
1 2 3
OutputCopy2NoteIn the first sample case there is only one sum 1+2=31+2=3.In the second sample case there are three sums: 1+2=31+2=3, 1+3=41+3=4, 2+3=52+3=5. In binary they are represented as 0112⊕1002⊕1012=01020112⊕1002⊕1012=0102, thus the answer is 2.⊕⊕ is the bitwise xor operation. To define x⊕yx⊕y, consider binary representations of integers xx and yy. We put the ii-th bit of the result to be 1 when exactly one of the ii-th bits of xx and yy is 1. Otherwise, the ii-th bit of the result is put to be 0. For example, 01012⊕00112=0110201012⊕00112=01102. | [
"binary search",
"bitmasks",
"constructive algorithms",
"data structures",
"math",
"sortings"
] | import sys
input = sys.stdin.buffer.readline
from functools import reduce
def __flatten(A):
return reduce(lambda x, y: x+y, A)
def radix(a):
digits = len(str(max(a)))
for i in range (digits):
b = [[] for _ in range (10)]
e = pow(10, i)
for j in a:
num = (j//e)%10
b[num].append(j)
a *= 0
for l in b:
a += l
N = int(input())
a = list(map(int, input().split()))
res = 0
radix(a)
for k in range(max(a).bit_length(), -1, -1):
z = 1 << k
c = 0
j = jj = jjj = N - 1
for i in range(N):
while j >= 0 and a[i] + a[j] >= z: j -= 1
while jj >= 0 and a[i] + a[jj] >= z << 1: jj -= 1
while jjj >= 0 and a[i] + a[jjj] >= z * 3: jjj -= 1
c += N - 1 - max(i, j) + max(i, jj) - max(i, jjj)
res += z * (c & 1)
for i in range(N): a[i] = min(a[i] ^ z, a[i])
a.sort()
#print(res, a, c)
print(res) | py |
1304 | C | C. Air Conditionertime limit per test1 secondmemory limit per test256 megabytesinputstandard inputoutputstandard outputGildong owns a bulgogi restaurant. The restaurant has a lot of customers; so many of them like to make a reservation before visiting it.Gildong tries so hard to satisfy the customers that he even memorized all customers' preferred temperature ranges! Looking through the reservation list, he wants to satisfy all customers by controlling the temperature of the restaurant.The restaurant has an air conditioner that has 3 states: off, heating, and cooling. When it's off, the restaurant's temperature remains the same. When it's heating, the temperature increases by 1 in one minute. Lastly, when it's cooling, the temperature decreases by 1 in one minute. Gildong can change the state as many times as he wants, at any integer minutes. The air conditioner is off initially.Each customer is characterized by three values: titi — the time (in minutes) when the ii-th customer visits the restaurant, lili — the lower bound of their preferred temperature range, and hihi — the upper bound of their preferred temperature range.A customer is satisfied if the temperature is within the preferred range at the instant they visit the restaurant. Formally, the ii-th customer is satisfied if and only if the temperature is between lili and hihi (inclusive) in the titi-th minute.Given the initial temperature, the list of reserved customers' visit times and their preferred temperature ranges, you're going to help him find if it's possible to satisfy all customers.InputEach test contains one or more test cases. The first line contains the number of test cases qq (1≤q≤5001≤q≤500). Description of the test cases follows.The first line of each test case contains two integers nn and mm (1≤n≤1001≤n≤100, −109≤m≤109−109≤m≤109), where nn is the number of reserved customers and mm is the initial temperature of the restaurant.Next, nn lines follow. The ii-th line of them contains three integers titi, lili, and hihi (1≤ti≤1091≤ti≤109, −109≤li≤hi≤109−109≤li≤hi≤109), where titi is the time when the ii-th customer visits, lili is the lower bound of their preferred temperature range, and hihi is the upper bound of their preferred temperature range. The preferred temperature ranges are inclusive.The customers are given in non-decreasing order of their visit time, and the current time is 00.OutputFor each test case, print "YES" if it is possible to satisfy all customers. Otherwise, print "NO".You can print each letter in any case (upper or lower).ExampleInputCopy4
3 0
5 1 2
7 3 5
10 -1 0
2 12
5 7 10
10 16 20
3 -100
100 0 0
100 -50 50
200 100 100
1 100
99 -100 0
OutputCopyYES
NO
YES
NO
NoteIn the first case; Gildong can control the air conditioner to satisfy all customers in the following way: At 00-th minute, change the state to heating (the temperature is 0). At 22-nd minute, change the state to off (the temperature is 2). At 55-th minute, change the state to heating (the temperature is 2, the 11-st customer is satisfied). At 66-th minute, change the state to off (the temperature is 3). At 77-th minute, change the state to cooling (the temperature is 3, the 22-nd customer is satisfied). At 1010-th minute, the temperature will be 0, which satisfies the last customer. In the third case, Gildong can change the state to heating at 00-th minute and leave it be. Then all customers will be satisfied. Note that the 11-st customer's visit time equals the 22-nd customer's visit time.In the second and the fourth case, Gildong has to make at least one customer unsatisfied. | [
"dp",
"greedy",
"implementation",
"sortings",
"two pointers"
] | # chalo shuru karte hai!
import sys
from math import *
from itertools import *
from heapq import heapify, heappop, heappush
from bisect import bisect, bisect_left, bisect_right
from collections import deque, Counter, defaultdict as dd
mod = 10**9+7
# Sangeeta Singh
def input(): return sys.stdin.readline().rstrip("\r\n")
def ri(): return int(input())
def rl(): return list(map(int, input().split()))
def rls(): return list(map(str, input().split()))
def isPowerOfTwo(x): return (x and (not(x & (x - 1))))
def lcm(x, y): return (x*y)//gcd(x, y)
def alpha(x): return ord(x)-ord('A')
def gcd(x, y):
while y:
x, y = y, x % y
return x
def d2b(n):
s = bin(n).replace("0b", "")
return (34-len(s))*'0'+s
def highestPowerof2(x):
x |= x >> 1
x |= x >> 2
x |= x >> 4
x |= x >> 8
x |= x >> 16
return x ^ (x >> 1)
def isPrime(x):
if x == 1:
return False
if x == 2:
return True
for i in range(2, int(x ** 0.5) + 1):
if x % i == 0:
return False
return True
def factors(n):
i = 1
ans = []
while i <= sqrt(n):
if (n % i == 0):
if (n / i != i):
ans.append(int(n/i))
if i != 1:
ans.append(i)
i += 1
return ans
############ Main! #############
Primes = [1] * 100001
primeNos = []
def SieveOfEratosthenes(n):
p = 2
while (p * p <= n):
if (Primes[p]):
for i in range(p * p, n+1, p):
Primes[i] = 0
p += 1
for p in range(2, n+1):
if Primes[p]:
primeNos.append(p)
# SieveOfEratosthenes(100000)
# P = len(primeNos)
# print("P", P)
# for _ in range(1):
for _ in range(ri()):
n, m = rl()
a = [rl() for i in range(n)]
a.sort()
mn = m
mx = m
prev = 0
for i in range(n):
k = a[i][0]-prev
mn = mn-k
mx = mx+k
if mn > a[i][2] or mx < a[i][1]:
print("NO")
break
mn = max(mn, a[i][1])
mx = min(mx, a[i][2])
prev = a[i][0]
else:
print("YES")
| py |
1285 | B | B. Just Eat It!time limit per test1 secondmemory limit per test256 megabytesinputstandard inputoutputstandard outputToday; Yasser and Adel are at the shop buying cupcakes. There are nn cupcake types, arranged from 11 to nn on the shelf, and there are infinitely many of each type. The tastiness of a cupcake of type ii is an integer aiai. There are both tasty and nasty cupcakes, so the tastiness can be positive, zero or negative.Yasser, of course, wants to try them all, so he will buy exactly one cupcake of each type.On the other hand, Adel will choose some segment [l,r][l,r] (1≤l≤r≤n)(1≤l≤r≤n) that does not include all of cupcakes (he can't choose [l,r]=[1,n][l,r]=[1,n]) and buy exactly one cupcake of each of types l,l+1,…,rl,l+1,…,r.After that they will compare the total tastiness of the cupcakes each of them have bought. Yasser will be happy if the total tastiness of cupcakes he buys is strictly greater than the total tastiness of cupcakes Adel buys regardless of Adel's choice.For example, let the tastinesses of the cupcakes be [7,4,−1][7,4,−1]. Yasser will buy all of them, the total tastiness will be 7+4−1=107+4−1=10. Adel can choose segments [7],[4],[−1],[7,4][7],[4],[−1],[7,4] or [4,−1][4,−1], their total tastinesses are 7,4,−1,117,4,−1,11 and 33, respectively. Adel can choose segment with tastiness 1111, and as 1010 is not strictly greater than 1111, Yasser won't be happy :(Find out if Yasser will be happy after visiting the shop.InputEach test contains multiple test cases. The first line contains the number of test cases tt (1≤t≤1041≤t≤104). The description of the test cases follows.The first line of each test case contains nn (2≤n≤1052≤n≤105).The second line of each test case contains nn integers a1,a2,…,ana1,a2,…,an (−109≤ai≤109−109≤ai≤109), where aiai represents the tastiness of the ii-th type of cupcake.It is guaranteed that the sum of nn over all test cases doesn't exceed 105105.OutputFor each test case, print "YES", if the total tastiness of cupcakes Yasser buys will always be strictly greater than the total tastiness of cupcakes Adel buys regardless of Adel's choice. Otherwise, print "NO".ExampleInputCopy3
4
1 2 3 4
3
7 4 -1
3
5 -5 5
OutputCopyYES
NO
NO
NoteIn the first example; the total tastiness of any segment Adel can choose is less than the total tastiness of all cupcakes.In the second example; Adel will choose the segment [1,2][1,2] with total tastiness 1111, which is not less than the total tastiness of all cupcakes, which is 1010.In the third example, Adel can choose the segment [3,3][3,3] with total tastiness of 55. Note that Yasser's cupcakes' total tastiness is also 55, so in that case, the total tastiness of Yasser's cupcakes isn't strictly greater than the total tastiness of Adel's cupcakes. | [
"dp",
"greedy",
"implementation"
] | import math as mt
from collections import defaultdict,deque
import sys
from bisect import bisect_right as b_r
from bisect import bisect_left as b_l
# from os import path
# from heapq import *
mod=1000000007
INT_MAX = sys.maxsize-1
INT_MIN = -sys.maxsize
# if(path.exists('inputt.txt')):
# sys.stdin = open('inputt.txt','r')
# sys.stdout = open('output.txt','w')
# else:
# # input=sys.stdin.readline
# pass
input = lambda: sys.stdin.readline().rstrip("\r\n")
def myyy__answer():
n=int(input())
a=list(map(int,input().split()))
ans=INT_MIN;temp=0
for i in range(n-1):
ans=max(ans,temp+a[i])
temp+=a[i]
if(temp<0):
temp=0
temp=0
for i in range(1,n):
ans=max(ans,temp+a[i])
temp+=a[i]
if(temp<0):
temp=0
# print(sum(a),ans)
if(sum(a)>ans):
print("YES")
else:
print("NO")
if __name__ == "__main__":
for _ in range(int(input())):
# print(myyy__answer())
myyy__answer() | py |
1291 | B | B. Array Sharpeningtime limit per test1 secondmemory limit per test256 megabytesinputstandard inputoutputstandard outputYou're given an array a1,…,ana1,…,an of nn non-negative integers.Let's call it sharpened if and only if there exists an integer 1≤k≤n1≤k≤n such that a1<a2<…<aka1<a2<…<ak and ak>ak+1>…>anak>ak+1>…>an. In particular, any strictly increasing or strictly decreasing array is sharpened. For example: The arrays [4][4], [0,1][0,1], [12,10,8][12,10,8] and [3,11,15,9,7,4][3,11,15,9,7,4] are sharpened; The arrays [2,8,2,8,6,5][2,8,2,8,6,5], [0,1,1,0][0,1,1,0] and [2,5,6,9,8,8][2,5,6,9,8,8] are not sharpened. You can do the following operation as many times as you want: choose any strictly positive element of the array, and decrease it by one. Formally, you can choose any ii (1≤i≤n1≤i≤n) such that ai>0ai>0 and assign ai:=ai−1ai:=ai−1.Tell if it's possible to make the given array sharpened using some number (possibly zero) of these operations.InputThe input consists of multiple test cases. The first line contains a single integer tt (1≤t≤15 0001≤t≤15 000) — the number of test cases. The description of the test cases follows.The first line of each test case contains a single integer nn (1≤n≤3⋅1051≤n≤3⋅105).The second line of each test case contains a sequence of nn non-negative integers a1,…,ana1,…,an (0≤ai≤1090≤ai≤109).It is guaranteed that the sum of nn over all test cases does not exceed 3⋅1053⋅105.OutputFor each test case, output a single line containing "Yes" (without quotes) if it's possible to make the given array sharpened using the described operations, or "No" (without quotes) otherwise.ExampleInputCopy10
1
248618
3
12 10 8
6
100 11 15 9 7 8
4
0 1 1 0
2
0 0
2
0 1
2
1 0
2
1 1
3
0 1 0
3
1 0 1
OutputCopyYes
Yes
Yes
No
No
Yes
Yes
Yes
Yes
No
NoteIn the first and the second test case of the first test; the given array is already sharpened.In the third test case of the first test; we can transform the array into [3,11,15,9,7,4][3,11,15,9,7,4] (decrease the first element 9797 times and decrease the last element 44 times). It is sharpened because 3<11<153<11<15 and 15>9>7>415>9>7>4.In the fourth test case of the first test, it's impossible to make the given array sharpened. | [
"greedy",
"implementation"
] | for _ in range(int(input())):
n=int(input())
arr=list(map(int,input().split(" ")))
if n%2:
yes=1
for i in range(n):
if arr[i]<min(i,n-1-i):
yes=0
break
print("yes" if yes else "no")
else:
yes=1
for i in range(n):
if arr[i]<min(i,n-1-i):
yes=0
break
if arr[n//2]==n//2-1 and arr[n//2-1]==n//2-1:
yes=0
print("yes" if yes else "no") | py |
1294 | A | A. Collecting Coinstime limit per test2 secondsmemory limit per test256 megabytesinputstandard inputoutputstandard outputPolycarp has three sisters: Alice; Barbara, and Cerene. They're collecting coins. Currently, Alice has aa coins, Barbara has bb coins and Cerene has cc coins. Recently Polycarp has returned from the trip around the world and brought nn coins.He wants to distribute all these nn coins between his sisters in such a way that the number of coins Alice has is equal to the number of coins Barbara has and is equal to the number of coins Cerene has. In other words, if Polycarp gives AA coins to Alice, BB coins to Barbara and CC coins to Cerene (A+B+C=nA+B+C=n), then a+A=b+B=c+Ca+A=b+B=c+C.Note that A, B or C (the number of coins Polycarp gives to Alice, Barbara and Cerene correspondingly) can be 0.Your task is to find out if it is possible to distribute all nn coins between sisters in a way described above.You have to answer tt independent test cases.InputThe first line of the input contains one integer tt (1≤t≤1041≤t≤104) — the number of test cases.The next tt lines describe test cases. Each test case is given on a new line and consists of four space-separated integers a,b,ca,b,c and nn (1≤a,b,c,n≤1081≤a,b,c,n≤108) — the number of coins Alice has, the number of coins Barbara has, the number of coins Cerene has and the number of coins Polycarp has.OutputFor each test case, print "YES" if Polycarp can distribute all nn coins between his sisters and "NO" otherwise.ExampleInputCopy5
5 3 2 8
100 101 102 105
3 2 1 100000000
10 20 15 14
101 101 101 3
OutputCopyYES
YES
NO
NO
YES
| [
"math"
] | n = int(input())
for i in range(n):
a,b,c,n = [int(i) for i in input().split()]
z = max(a,b,c)
p = n - (z-a + z-b + z-c)
if p>=0:
if p%3==0:
print("YES")
else:
print("NO")
if p<0:
print("NO") | py |
1316 | D | D. Nash Matrixtime limit per test2 secondsmemory limit per test256 megabytesinputstandard inputoutputstandard outputNash designed an interesting yet simple board game where a player is simply required to follow instructions written on the cell where the player currently stands. This board game is played on the n×nn×n board. Rows and columns of this board are numbered from 11 to nn. The cell on the intersection of the rr-th row and cc-th column is denoted by (r,c)(r,c).Some cells on the board are called blocked zones. On each cell of the board, there is written one of the following 55 characters — UU, DD, LL, RR or XX — instructions for the player. Suppose that the current cell is (r,c)(r,c). If the character is RR, the player should move to the right cell (r,c+1)(r,c+1), for LL the player should move to the left cell (r,c−1)(r,c−1), for UU the player should move to the top cell (r−1,c)(r−1,c), for DD the player should move to the bottom cell (r+1,c)(r+1,c). Finally, if the character in the cell is XX, then this cell is the blocked zone. The player should remain in this cell (the game for him isn't very interesting from now on).It is guaranteed that the characters are written in a way that the player will never have to step outside of the board, no matter at which cell he starts.As a player starts from a cell, he moves according to the character in the current cell. The player keeps moving until he lands in a blocked zone. It is also possible that the player will keep moving infinitely long.For every of the n2n2 cells of the board Alice, your friend, wants to know, how will the game go, if the player starts in this cell. For each starting cell of the board, she writes down the cell that the player stops at, or that the player never stops at all. She gives you the information she has written: for each cell (r,c)(r,c) she wrote: a pair (xx,yy), meaning if a player had started at (r,c)(r,c), he would end up at cell (xx,yy). or a pair (−1−1,−1−1), meaning if a player had started at (r,c)(r,c), he would keep moving infinitely long and would never enter the blocked zone. It might be possible that Alice is trying to fool you and there's no possible grid that satisfies all the constraints Alice gave you. For the given information Alice provided you, you are required to decipher a possible board, or to determine that such a board doesn't exist. If there exist several different boards that satisfy the provided information, you can find any of them.InputThe first line of the input contains a single integer nn (1≤n≤1031≤n≤103) — the side of the board.The ii-th of the next nn lines of the input contains 2n2n integers x1,y1,x2,y2,…,xn,ynx1,y1,x2,y2,…,xn,yn, where (xj,yj)(xj,yj) (1≤xj≤n,1≤yj≤n1≤xj≤n,1≤yj≤n, or (xj,yj)=(−1,−1)(xj,yj)=(−1,−1)) is the pair written by Alice for the cell (i,j)(i,j). OutputIf there doesn't exist a board satisfying the information that Alice gave you, print a single line containing INVALID. Otherwise, in the first line print VALID. In the ii-th of the next nn lines, print the string of nn characters, corresponding to the characters in the ii-th row of the suitable board you found. Each character of a string can either be UU, DD, LL, RR or XX. If there exist several different boards that satisfy the provided information, you can find any of them.ExamplesInputCopy2
1 1 1 1
2 2 2 2
OutputCopyVALID
XL
RX
InputCopy3
-1 -1 -1 -1 -1 -1
-1 -1 2 2 -1 -1
-1 -1 -1 -1 -1 -1
OutputCopyVALID
RRD
UXD
ULLNoteFor the sample test 1 :The given grid in output is a valid one. If the player starts at (1,1)(1,1), he doesn't move any further following XX and stops there. If the player starts at (1,2)(1,2), he moves to left following LL and stops at (1,1)(1,1). If the player starts at (2,1)(2,1), he moves to right following RR and stops at (2,2)(2,2). If the player starts at (2,2)(2,2), he doesn't move any further following XX and stops there. The simulation can be seen below : For the sample test 2 : The given grid in output is a valid one, as a player starting at any cell other than the one at center (2,2)(2,2), keeps moving in an infinitely long cycle and never stops. Had he started at (2,2)(2,2), he wouldn't have moved further following instruction XX .The simulation can be seen below : | [
"constructive algorithms",
"dfs and similar",
"graphs",
"implementation"
] | from math import sqrt
import sys
from functools import reduce
from collections import deque
import io
import os
# input = io.BytesIO(os.read(0, os.fstat(0).st_size)).readline
def input():
return sys.stdin.readline().strip()
def iinput():
return int(input())
def tinput():
return input().split()
def rinput():
return map(int, tinput())
def rlinput():
return list(rinput())
mod = int(1e9)+7
def factors(n):
return set(reduce(list.__add__,
([i, n//i] for i in range(1, int(n**0.5) + 1) if n % i == 0)))
# ----------------------------------------------------
if __name__ == "__main__":
n = iinput()
mat = [[[-1, -1] for i in range(n)]for i in range(n)]
ans = [['' for i in range(n)]for i in range(n)]
stack = deque()
for i in range(n):
temp = rlinput()
for j in range(n):
mat[i][j][0] = temp[2*j]
mat[i][j][1] = temp[2*j+1]
if (temp[2*j], temp[2*j+1]) == (i+1, j+1):
ans[i][j] = 'X'
stack.append((i, j))
# Part 2
while stack:
state = stack.pop()
i, j = state
if i < n-1 and ans[i+1][j] == '':
if mat[i+1][j] == mat[i][j]:
ans[i+1][j] = 'U'
stack.append((i+1, j))
if j < n-1 and ans[i][j+1] == '':
if mat[i][j+1] == mat[i][j]:
ans[i][j+1] = 'L'
stack.append((i, j+1))
if i > 0 and ans[i-1][j] == '':
if mat[i-1][j] == mat[i][j]:
ans[i-1][j] = 'D'
stack.append((i-1, j))
if j > 0 and ans[i][j-1] == '':
if mat[i][j-1] == mat[i][j]:
ans[i][j-1] = 'R'
stack.append((i, j-1))
for i in range(n):
for j in range(n):
if mat[i][j] == [-1, -1]:
if i < n-1 and mat[i+1][j] == [-1, -1]:
ans[i][j] = 'D'
continue
if j < n-1 and mat[i][j+1] == [-1, -1]:
ans[i][j] = 'R'
continue
if i > 0 and mat[i-1][j] == [-1, -1]:
ans[i][j] = 'U'
continue
if j > 0 and mat[i][j-1] == [-1, -1]:
ans[i][j] = 'L'
continue
flag = True
for i in ans:
for j in i:
if j == '':
print('INVALID')
sys.exit()
print('VALID')
for i in ans:
print(''.join(i))
| py |
1294 | B | B. Collecting Packagestime limit per test1 secondmemory limit per test256 megabytesinputstandard inputoutputstandard outputThere is a robot in a warehouse and nn packages he wants to collect. The warehouse can be represented as a coordinate grid. Initially, the robot stays at the point (0,0)(0,0). The ii-th package is at the point (xi,yi)(xi,yi). It is guaranteed that there are no two packages at the same point. It is also guaranteed that the point (0,0)(0,0) doesn't contain a package.The robot is semi-broken and only can move up ('U') and right ('R'). In other words, in one move the robot can go from the point (x,y)(x,y) to the point (x+1,yx+1,y) or to the point (x,y+1)(x,y+1).As we say above, the robot wants to collect all nn packages (in arbitrary order). He wants to do it with the minimum possible number of moves. If there are several possible traversals, the robot wants to choose the lexicographically smallest path.The string ss of length nn is lexicographically less than the string tt of length nn if there is some index 1≤j≤n1≤j≤n that for all ii from 11 to j−1j−1 si=tisi=ti and sj<tjsj<tj. It is the standard comparison of string, like in a dictionary. Most programming languages compare strings in this way.InputThe first line of the input contains an integer tt (1≤t≤1001≤t≤100) — the number of test cases. Then test cases follow.The first line of a test case contains one integer nn (1≤n≤10001≤n≤1000) — the number of packages.The next nn lines contain descriptions of packages. The ii-th package is given as two integers xixi and yiyi (0≤xi,yi≤10000≤xi,yi≤1000) — the xx-coordinate of the package and the yy-coordinate of the package.It is guaranteed that there are no two packages at the same point. It is also guaranteed that the point (0,0)(0,0) doesn't contain a package.The sum of all values nn over test cases in the test doesn't exceed 10001000.OutputPrint the answer for each test case.If it is impossible to collect all nn packages in some order starting from (0,00,0), print "NO" on the first line.Otherwise, print "YES" in the first line. Then print the shortest path — a string consisting of characters 'R' and 'U'. Among all such paths choose the lexicographically smallest path.Note that in this problem "YES" and "NO" can be only uppercase words, i.e. "Yes", "no" and "YeS" are not acceptable.ExampleInputCopy3
5
1 3
1 2
3 3
5 5
4 3
2
1 0
0 1
1
4 3
OutputCopyYES
RUUURRRRUU
NO
YES
RRRRUUU
NoteFor the first test case in the example the optimal path RUUURRRRUU is shown below: | [
"implementation",
"sortings"
] | for _ in range(int(input())):
n = int(input())
cords = []
for i in range(n):
a, b = map(int, input().split())
cords.append([a, b])
cords = sorted(cords, key = lambda t: t[0])
cords = sorted(cords, key = lambda t: t[1])
prevx = 0
prevy = 0
ans = ''
flag = False
for a, b in cords:
if a < prevx:
flag = True
break
xcord = a - prevx
ycord = b - prevy
ans += xcord*"R" + ycord*"U"
prevx = a
prevy = b
if flag:
print('NO')
else:
print('YES')
print(ans) | py |
1305 | B | B. Kuroni and Simple Stringstime limit per test1 secondmemory limit per test256 megabytesinputstandard inputoutputstandard outputNow that Kuroni has reached 10 years old; he is a big boy and doesn't like arrays of integers as presents anymore. This year he wants a Bracket sequence as a Birthday present. More specifically, he wants a bracket sequence so complex that no matter how hard he tries, he will not be able to remove a simple subsequence!We say that a string formed by nn characters '(' or ')' is simple if its length nn is even and positive, its first n2n2 characters are '(', and its last n2n2 characters are ')'. For example, the strings () and (()) are simple, while the strings )( and ()() are not simple.Kuroni will be given a string formed by characters '(' and ')' (the given string is not necessarily simple). An operation consists of choosing a subsequence of the characters of the string that forms a simple string and removing all the characters of this subsequence from the string. Note that this subsequence doesn't have to be continuous. For example, he can apply the operation to the string ')()(()))', to choose a subsequence of bold characters, as it forms a simple string '(())', delete these bold characters from the string and to get '))()'. Kuroni has to perform the minimum possible number of operations on the string, in such a way that no more operations can be performed on the remaining string. The resulting string does not have to be empty.Since the given string is too large, Kuroni is unable to figure out how to minimize the number of operations. Can you help him do it instead?A sequence of characters aa is a subsequence of a string bb if aa can be obtained from bb by deletion of several (possibly, zero or all) characters.InputThe only line of input contains a string ss (1≤|s|≤10001≤|s|≤1000) formed by characters '(' and ')', where |s||s| is the length of ss.OutputIn the first line, print an integer kk — the minimum number of operations you have to apply. Then, print 2k2k lines describing the operations in the following format:For each operation, print a line containing an integer mm — the number of characters in the subsequence you will remove.Then, print a line containing mm integers 1≤a1<a2<⋯<am1≤a1<a2<⋯<am — the indices of the characters you will remove. All integers must be less than or equal to the length of the current string, and the corresponding subsequence must form a simple string.If there are multiple valid sequences of operations with the smallest kk, you may print any of them.ExamplesInputCopy(()((
OutputCopy1
2
1 3
InputCopy)(
OutputCopy0
InputCopy(()())
OutputCopy1
4
1 2 5 6
NoteIn the first sample; the string is '(()(('. The operation described corresponds to deleting the bolded subsequence. The resulting string is '((('; and no more operations can be performed on it. Another valid answer is choosing indices 22 and 33, which results in the same final string.In the second sample, it is already impossible to perform any operations. | [
"constructive algorithms",
"greedy",
"strings",
"two pointers"
] | def min_opr(s):
min_oprs = 0
output = []
# While I can make an operation
op = True
while op:
op = False
# Indexes to remove from the string
to_remove = set()
# Two pointers
a = 0
b = len(s) - 1
while a < b:
# Find the two parentheses
if s[a] == "(":
while s[b] != ")" and b > 0:
b -= 1
if a < b:
# These two indexes are include in a good string
to_remove.add(a)
to_remove.add(b)
b -= 1
a += 1
if len(to_remove) > 0:
op = True
output.append(
(len(to_remove), [str(i + 1) for i in sorted(list(to_remove))])
)
s = "".join([c for i, c in enumerate(s) if i not in to_remove])
min_oprs += 1
# Print the answer
return output
def solve():
s = input().strip()
result = min_opr(s)
print(len(result))
for r in result:
print(r[0])
print(*r[1])
solve()
| py |
1141 | F1 | F1. Same Sum Blocks (Easy)time limit per test2 secondsmemory limit per test256 megabytesinputstandard inputoutputstandard outputThis problem is given in two editions; which differ exclusively in the constraints on the number nn.You are given an array of integers a[1],a[2],…,a[n].a[1],a[2],…,a[n]. A block is a sequence of contiguous (consecutive) elements a[l],a[l+1],…,a[r]a[l],a[l+1],…,a[r] (1≤l≤r≤n1≤l≤r≤n). Thus, a block is defined by a pair of indices (l,r)(l,r).Find a set of blocks (l1,r1),(l2,r2),…,(lk,rk)(l1,r1),(l2,r2),…,(lk,rk) such that: They do not intersect (i.e. they are disjoint). Formally, for each pair of blocks (li,ri)(li,ri) and (lj,rj(lj,rj) where i≠ji≠j either ri<ljri<lj or rj<lirj<li. For each block the sum of its elements is the same. Formally, a[l1]+a[l1+1]+⋯+a[r1]=a[l2]+a[l2+1]+⋯+a[r2]=a[l1]+a[l1+1]+⋯+a[r1]=a[l2]+a[l2+1]+⋯+a[r2]= ⋯=⋯= a[lk]+a[lk+1]+⋯+a[rk].a[lk]+a[lk+1]+⋯+a[rk]. The number of the blocks in the set is maximum. Formally, there does not exist a set of blocks (l′1,r′1),(l′2,r′2),…,(l′k′,r′k′)(l1′,r1′),(l2′,r2′),…,(lk′′,rk′′) satisfying the above two requirements with k′>kk′>k. The picture corresponds to the first example. Blue boxes illustrate blocks. Write a program to find such a set of blocks.InputThe first line contains integer nn (1≤n≤501≤n≤50) — the length of the given array. The second line contains the sequence of elements a[1],a[2],…,a[n]a[1],a[2],…,a[n] (−105≤ai≤105−105≤ai≤105).OutputIn the first line print the integer kk (1≤k≤n1≤k≤n). The following kk lines should contain blocks, one per line. In each line print a pair of indices li,rili,ri (1≤li≤ri≤n1≤li≤ri≤n) — the bounds of the ii-th block. You can print blocks in any order. If there are multiple answers, print any of them.ExamplesInputCopy7
4 1 2 2 1 5 3
OutputCopy3
7 7
2 3
4 5
InputCopy11
-5 -4 -3 -2 -1 0 1 2 3 4 5
OutputCopy2
3 4
1 1
InputCopy4
1 1 1 1
OutputCopy4
4 4
1 1
2 2
3 3
| [
"greedy"
] | import re
import functools
import random
import sys
import os
import math
from collections import Counter, defaultdict, deque
from functools import lru_cache, reduce, cmp_to_key
from itertools import accumulate, combinations, permutations
from heapq import nsmallest, nlargest, heappushpop, heapify, heappop, heappush
from io import BytesIO, IOBase
from copy import deepcopy
import threading
from typing import *
from operator import add, xor, mul, ior, iand, itemgetter
import bisect
from string import ascii_lowercase, ascii_uppercase
BUFSIZE = 4096
inf = float('inf')
if "PyPy" in sys.version:
import pypyjit; pypyjit.set_param('max_unroll_recursion=-1')
class FastIO(IOBase):
newlines = 0
def __init__(self, file):
self._fd = file.fileno()
self.buffer = BytesIO()
self.writable = "x" in file.mode or "r" not in file.mode
self.write = self.buffer.write if self.writable else None
def read(self):
while True:
b = os.read(self._fd, max(os.fstat(self._fd).st_size, BUFSIZE))
if not b:
break
ptr = self.buffer.tell()
self.buffer.seek(0, 2), self.buffer.write(b), self.buffer.seek(ptr)
self.newlines = 0
return self.buffer.read()
def readline(self):
while self.newlines == 0:
b = os.read(self._fd, max(os.fstat(self._fd).st_size, BUFSIZE))
self.newlines = b.count(b"\n") + (not b)
ptr = self.buffer.tell()
self.buffer.seek(0, 2), self.buffer.write(b), self.buffer.seek(ptr)
self.newlines -= 1
return self.buffer.readline()
def flush(self):
if self.writable:
os.write(self._fd, self.buffer.getvalue())
self.buffer.truncate(0), self.buffer.seek(0)
class IOWrapper(IOBase):
def __init__(self, file):
self.buffer = FastIO(file)
self.flush = self.buffer.flush
self.writable = self.buffer.writable
self.write = lambda s: self.buffer.write(s.encode("ascii"))
self.read = lambda: self.buffer.read().decode("ascii")
self.readline = lambda: self.buffer.readline().decode("ascii")
sys.stdin = IOWrapper(sys.stdin)
sys.stdout = IOWrapper(sys.stdout)
input = lambda: sys.stdin.readline().rstrip("\r\n")
def I():
return input()
def II():
return int(input())
def MII():
return map(int, input().split())
def LI():
return list(input().split())
def LII():
return list(map(int, input().split()))
def GMI():
return map(lambda x: int(x) - 1, input().split())
def LGMI():
return list(map(lambda x: int(x) - 1, input().split()))
from types import GeneratorType
def bootstrap(f, stack=[]):
def wrappedfunc(*args, **kwargs):
if stack:
return f(*args, **kwargs)
else:
to = f(*args, **kwargs)
while True:
if type(to) is GeneratorType:
stack.append(to)
to = next(to)
else:
stack.pop()
if not stack:
break
to = stack[-1].send(to)
return to
return wrappedfunc
n = II()
nums = LII()
pre, rec = list(accumulate(nums, initial=0)), defaultdict(list)
for i in range(n):
for j in range(i + 1):
rec[pre[i+1] - pre[j]].append((j+1, i+1))
ans = []
for ls in rec.values():
if len(ls) <= len(ans): continue
tmp = [ls[0]]
for i in range(1, len(ls)):
if ls[i][0] <= tmp[-1][1]:
continue
tmp.append(ls[i])
if len(tmp) > len(ans):
ans = tmp
print(len(ans))
for l, r in ans:
print(l, r) | py |
1293 | A | A. ConneR and the A.R.C. Markland-Ntime limit per test1 secondmemory limit per test256 megabytesinputstandard inputoutputstandard outputSakuzyo - ImprintingA.R.C. Markland-N is a tall building with nn floors numbered from 11 to nn. Between each two adjacent floors in the building, there is a staircase connecting them.It's lunchtime for our sensei Colin "ConneR" Neumann Jr, and he's planning for a location to enjoy his meal.ConneR's office is at floor ss of the building. On each floor (including floor ss, of course), there is a restaurant offering meals. However, due to renovations being in progress, kk of the restaurants are currently closed, and as a result, ConneR can't enjoy his lunch there.CooneR wants to reach a restaurant as quickly as possible to save time. What is the minimum number of staircases he needs to walk to reach a closest currently open restaurant.Please answer him quickly, and you might earn his praise and even enjoy the lunch with him in the elegant Neumanns' way!InputThe first line contains one integer tt (1≤t≤10001≤t≤1000) — the number of test cases in the test. Then the descriptions of tt test cases follow.The first line of a test case contains three integers nn, ss and kk (2≤n≤1092≤n≤109, 1≤s≤n1≤s≤n, 1≤k≤min(n−1,1000)1≤k≤min(n−1,1000)) — respectively the number of floors of A.R.C. Markland-N, the floor where ConneR is in, and the number of closed restaurants.The second line of a test case contains kk distinct integers a1,a2,…,aka1,a2,…,ak (1≤ai≤n1≤ai≤n) — the floor numbers of the currently closed restaurants.It is guaranteed that the sum of kk over all test cases does not exceed 10001000.OutputFor each test case print a single integer — the minimum number of staircases required for ConneR to walk from the floor ss to a floor with an open restaurant.ExampleInputCopy5
5 2 3
1 2 3
4 3 3
4 1 2
10 2 6
1 2 3 4 5 7
2 1 1
2
100 76 8
76 75 36 67 41 74 10 77
OutputCopy2
0
4
0
2
NoteIn the first example test case; the nearest floor with an open restaurant would be the floor 44.In the second example test case, the floor with ConneR's office still has an open restaurant, so Sensei won't have to go anywhere.In the third example test case, the closest open restaurant is on the 66-th floor. | [
"binary search",
"brute force",
"implementation"
] | # LUOGU_RID: 101845080
for _ in range(int(input())):
n, s, k = map(int, input().split())
a = list(map(int, input().split()))
c = [1] + [(s - i < 1) + (s + i > n) for i in range(1, k)]
for x in a:
if abs(x - s) < k:
c[abs(x - s)] += 1
p = 0
while p < k and c[p] == 2:
p += 1
print(p) | py |
1288 | A | A. Deadlinetime limit per test2 secondsmemory limit per test256 megabytesinputstandard inputoutputstandard outputAdilbek was assigned to a special project. For Adilbek it means that he has nn days to run a special program and provide its results. But there is a problem: the program needs to run for dd days to calculate the results.Fortunately, Adilbek can optimize the program. If he spends xx (xx is a non-negative integer) days optimizing the program, he will make the program run in ⌈dx+1⌉⌈dx+1⌉ days (⌈a⌉⌈a⌉ is the ceiling function: ⌈2.4⌉=3⌈2.4⌉=3, ⌈2⌉=2⌈2⌉=2). The program cannot be run and optimized simultaneously, so the total number of days he will spend is equal to x+⌈dx+1⌉x+⌈dx+1⌉.Will Adilbek be able to provide the generated results in no more than nn days?InputThe first line contains a single integer TT (1≤T≤501≤T≤50) — the number of test cases.The next TT lines contain test cases – one per line. Each line contains two integers nn and dd (1≤n≤1091≤n≤109, 1≤d≤1091≤d≤109) — the number of days before the deadline and the number of days the program runs.OutputPrint TT answers — one per test case. For each test case print YES (case insensitive) if Adilbek can fit in nn days or NO (case insensitive) otherwise.ExampleInputCopy3
1 1
4 5
5 11
OutputCopyYES
YES
NO
NoteIn the first test case; Adilbek decides not to optimize the program at all; since d≤nd≤n.In the second test case, Adilbek can spend 11 day optimizing the program and it will run ⌈52⌉=3⌈52⌉=3 days. In total, he will spend 44 days and will fit in the limit.In the third test case, it's impossible to fit in the limit. For example, if Adilbek will optimize the program 22 days, it'll still work ⌈112+1⌉=4⌈112+1⌉=4 days. | [
"binary search",
"brute force",
"math",
"ternary search"
] | import math
t = int(input())
result = []
for i in range(t):
nd = list(map(int, input().split()))
n = nd[0]
d = nd[-1]
flag = 0
if (d <= n):
result.append("YES")
continue
for i in range(n):
if (math.ceil(i + (d/(i+1))) <= n):
flag = 1
break
if (flag == 1):
result.append("YES")
else:
result.append("NO")
for i in result:
print(i)
| py |
1316 | A | A. Grade Allocationtime limit per test1 secondmemory limit per test256 megabytesinputstandard inputoutputstandard outputnn students are taking an exam. The highest possible score at this exam is mm. Let aiai be the score of the ii-th student. You have access to the school database which stores the results of all students.You can change each student's score as long as the following conditions are satisfied: All scores are integers 0≤ai≤m0≤ai≤m The average score of the class doesn't change. You are student 11 and you would like to maximize your own score.Find the highest possible score you can assign to yourself such that all conditions are satisfied.InputEach test contains multiple test cases. The first line contains the number of test cases tt (1≤t≤2001≤t≤200). The description of the test cases follows.The first line of each test case contains two integers nn and mm (1≤n≤1031≤n≤103, 1≤m≤1051≤m≤105) — the number of students and the highest possible score respectively.The second line of each testcase contains nn integers a1,a2,…,ana1,a2,…,an (0≤ai≤m0≤ai≤m) — scores of the students.OutputFor each testcase, output one integer — the highest possible score you can assign to yourself such that both conditions are satisfied._ExampleInputCopy2
4 10
1 2 3 4
4 5
1 2 3 4
OutputCopy10
5
NoteIn the first case; a=[1,2,3,4]a=[1,2,3,4], with average of 2.52.5. You can change array aa to [10,0,0,0][10,0,0,0]. Average remains 2.52.5, and all conditions are satisfied.In the second case, 0≤ai≤50≤ai≤5. You can change aa to [5,1,1,3][5,1,1,3]. You cannot increase a1a1 further as it will violate condition 0≤ai≤m0≤ai≤m. | [
"implementation"
] | # input the number of test cases
t = int(input())
# loop over range of t
for i in range(t):
# input the number of grades and max grade possible
n,m=str(input()).split(' ')
# input the list of grades
grades=str(input()).split(' ')
# turn the list of strings to ints because of this wretched python, sigh
grades=[int(grade) for grade in grades]
# get the sum of grades
sumOfGrades=sum(grades)
# now, if the sum of grades is greater than the max possible grade then we can only take the max grade possible
if sumOfGrades > int(m):
print(m)
# but if the class is sucking, we can take the sum because guess what we can't change the avg
# changing the avg means we can addd points from our pockets.
else:
print(sumOfGrades) | py |
1322 | A | A. Unusual Competitionstime limit per test1 secondmemory limit per test512 megabytesinputstandard inputoutputstandard outputA bracketed sequence is called correct (regular) if by inserting "+" and "1" you can get a well-formed mathematical expression from it. For example; sequences "(())()", "()" and "(()(()))" are correct, while ")(", "(()" and "(()))(" are not.The teacher gave Dmitry's class a very strange task — she asked every student to come up with a sequence of arbitrary length, consisting only of opening and closing brackets. After that all the students took turns naming the sequences they had invented. When Dima's turn came, he suddenly realized that all his classmates got the correct bracketed sequence, and whether he got the correct bracketed sequence, he did not know.Dima suspects now that he simply missed the word "correct" in the task statement, so now he wants to save the situation by modifying his sequence slightly. More precisely, he can the arbitrary number of times (possibly zero) perform the reorder operation.The reorder operation consists of choosing an arbitrary consecutive subsegment (substring) of the sequence and then reordering all the characters in it in an arbitrary way. Such operation takes ll nanoseconds, where ll is the length of the subsegment being reordered. It's easy to see that reorder operation doesn't change the number of opening and closing brackets. For example for "))((" he can choose the substring ")(" and do reorder ")()(" (this operation will take 22 nanoseconds).Since Dima will soon have to answer, he wants to make his sequence correct as fast as possible. Help him to do this, or determine that it's impossible.InputThe first line contains a single integer nn (1≤n≤1061≤n≤106) — the length of Dima's sequence.The second line contains string of length nn, consisting of characters "(" and ")" only.OutputPrint a single integer — the minimum number of nanoseconds to make the sequence correct or "-1" if it is impossible to do so.ExamplesInputCopy8
))((())(
OutputCopy6
InputCopy3
(()
OutputCopy-1
NoteIn the first example we can firstly reorder the segment from first to the fourth character; replacing it with "()()"; the whole sequence will be "()()())(". And then reorder the segment from the seventh to eighth character; replacing it with "()". In the end the sequence will be "()()()()"; while the total time spent is 4+2=64+2=6 nanoseconds. | [
"greedy"
] | n = int(input())
s = input()
ans = []
c = 0
for i in range(n):
c = c + 1 if s[i] == '(' else c - 1
ans.append(c)
c = 0
for i in range(n):
if(ans[i] < 0):
c += 1
elif(ans[i] == 0 and ans[i - 1] < 0):
c += 1
if(ans[-1] == 0):
print(c)
else:
print(-1)
| py |
1294 | E | E. Obtain a Permutationtime limit per test2 secondsmemory limit per test256 megabytesinputstandard inputoutputstandard outputYou are given a rectangular matrix of size n×mn×m consisting of integers from 11 to 2⋅1052⋅105.In one move, you can: choose any element of the matrix and change its value to any integer between 11 and n⋅mn⋅m, inclusive; take any column and shift it one cell up cyclically (see the example of such cyclic shift below). A cyclic shift is an operation such that you choose some jj (1≤j≤m1≤j≤m) and set a1,j:=a2,j,a2,j:=a3,j,…,an,j:=a1,ja1,j:=a2,j,a2,j:=a3,j,…,an,j:=a1,j simultaneously. Example of cyclic shift of the first column You want to perform the minimum number of moves to make this matrix look like this: In other words, the goal is to obtain the matrix, where a1,1=1,a1,2=2,…,a1,m=m,a2,1=m+1,a2,2=m+2,…,an,m=n⋅ma1,1=1,a1,2=2,…,a1,m=m,a2,1=m+1,a2,2=m+2,…,an,m=n⋅m (i.e. ai,j=(i−1)⋅m+jai,j=(i−1)⋅m+j) with the minimum number of moves performed.InputThe first line of the input contains two integers nn and mm (1≤n,m≤2⋅105,n⋅m≤2⋅1051≤n,m≤2⋅105,n⋅m≤2⋅105) — the size of the matrix.The next nn lines contain mm integers each. The number at the line ii and position jj is ai,jai,j (1≤ai,j≤2⋅1051≤ai,j≤2⋅105).OutputPrint one integer — the minimum number of moves required to obtain the matrix, where a1,1=1,a1,2=2,…,a1,m=m,a2,1=m+1,a2,2=m+2,…,an,m=n⋅ma1,1=1,a1,2=2,…,a1,m=m,a2,1=m+1,a2,2=m+2,…,an,m=n⋅m (ai,j=(i−1)m+jai,j=(i−1)m+j).ExamplesInputCopy3 3
3 2 1
1 2 3
4 5 6
OutputCopy6
InputCopy4 3
1 2 3
4 5 6
7 8 9
10 11 12
OutputCopy0
InputCopy3 4
1 6 3 4
5 10 7 8
9 2 11 12
OutputCopy2
NoteIn the first example; you can set a1,1:=7,a1,2:=8a1,1:=7,a1,2:=8 and a1,3:=9a1,3:=9 then shift the first, the second and the third columns cyclically, so the answer is 66. It can be shown that you cannot achieve a better answer.In the second example, the matrix is already good so the answer is 00.In the third example, it is enough to shift the second column cyclically twice to obtain a good matrix, so the answer is 22. | [
"greedy",
"implementation",
"math"
] | import sys
input=sys.stdin.readline
n,m=map(int,input().split())
a=[list(map(int,input().split())) for i in range(n)]
ans=0
for j in range(m):
idx={}
for i in range(n):
idx[i*m+(j+1)]=i
score_cyc=10**18
# up
r=0
cyc=[0]*n
for i in range(n):
if a[i][j] not in idx:
r+=1
else:
dist=i-idx[a[i][j]] if idx[a[i][j]]<=i else n+i-idx[a[i][j]]
cyc[dist%n]+=1
for i in range(n):
score_cyc=min(score_cyc,i+(n-cyc[i]-r)+r)
ans+=score_cyc
print(ans) | py |
1320 | B | B. Navigation Systemtime limit per test2 secondsmemory limit per test512 megabytesinputstandard inputoutputstandard outputThe map of Bertown can be represented as a set of nn intersections, numbered from 11 to nn and connected by mm one-way roads. It is possible to move along the roads from any intersection to any other intersection. The length of some path from one intersection to another is the number of roads that one has to traverse along the path. The shortest path from one intersection vv to another intersection uu is the path that starts in vv, ends in uu and has the minimum length among all such paths.Polycarp lives near the intersection ss and works in a building near the intersection tt. Every day he gets from ss to tt by car. Today he has chosen the following path to his workplace: p1p1, p2p2, ..., pkpk, where p1=sp1=s, pk=tpk=t, and all other elements of this sequence are the intermediate intersections, listed in the order Polycarp arrived at them. Polycarp never arrived at the same intersection twice, so all elements of this sequence are pairwise distinct. Note that you know Polycarp's path beforehand (it is fixed), and it is not necessarily one of the shortest paths from ss to tt.Polycarp's car has a complex navigation system installed in it. Let's describe how it works. When Polycarp starts his journey at the intersection ss, the system chooses some shortest path from ss to tt and shows it to Polycarp. Let's denote the next intersection in the chosen path as vv. If Polycarp chooses to drive along the road from ss to vv, then the navigator shows him the same shortest path (obviously, starting from vv as soon as he arrives at this intersection). However, if Polycarp chooses to drive to another intersection ww instead, the navigator rebuilds the path: as soon as Polycarp arrives at ww, the navigation system chooses some shortest path from ww to tt and shows it to Polycarp. The same process continues until Polycarp arrives at tt: if Polycarp moves along the road recommended by the system, it maintains the shortest path it has already built; but if Polycarp chooses some other path, the system rebuilds the path by the same rules.Here is an example. Suppose the map of Bertown looks as follows, and Polycarp drives along the path [1,2,3,4][1,2,3,4] (s=1s=1, t=4t=4): Check the picture by the link http://tk.codeforces.com/a.png When Polycarp starts at 11, the system chooses some shortest path from 11 to 44. There is only one such path, it is [1,5,4][1,5,4]; Polycarp chooses to drive to 22, which is not along the path chosen by the system. When Polycarp arrives at 22, the navigator rebuilds the path by choosing some shortest path from 22 to 44, for example, [2,6,4][2,6,4] (note that it could choose [2,3,4][2,3,4]); Polycarp chooses to drive to 33, which is not along the path chosen by the system. When Polycarp arrives at 33, the navigator rebuilds the path by choosing the only shortest path from 33 to 44, which is [3,4][3,4]; Polycarp arrives at 44 along the road chosen by the navigator, so the system does not have to rebuild anything. Overall, we get 22 rebuilds in this scenario. Note that if the system chose [2,3,4][2,3,4] instead of [2,6,4][2,6,4] during the second step, there would be only 11 rebuild (since Polycarp goes along the path, so the system maintains the path [3,4][3,4] during the third step).The example shows us that the number of rebuilds can differ even if the map of Bertown and the path chosen by Polycarp stays the same. Given this information (the map and Polycarp's path), can you determine the minimum and the maximum number of rebuilds that could have happened during the journey?InputThe first line contains two integers nn and mm (2≤n≤m≤2⋅1052≤n≤m≤2⋅105) — the number of intersections and one-way roads in Bertown, respectively.Then mm lines follow, each describing a road. Each line contains two integers uu and vv (1≤u,v≤n1≤u,v≤n, u≠vu≠v) denoting a road from intersection uu to intersection vv. All roads in Bertown are pairwise distinct, which means that each ordered pair (u,v)(u,v) appears at most once in these mm lines (but if there is a road (u,v)(u,v), the road (v,u)(v,u) can also appear).The following line contains one integer kk (2≤k≤n2≤k≤n) — the number of intersections in Polycarp's path from home to his workplace.The last line contains kk integers p1p1, p2p2, ..., pkpk (1≤pi≤n1≤pi≤n, all these integers are pairwise distinct) — the intersections along Polycarp's path in the order he arrived at them. p1p1 is the intersection where Polycarp lives (s=p1s=p1), and pkpk is the intersection where Polycarp's workplace is situated (t=pkt=pk). It is guaranteed that for every i∈[1,k−1]i∈[1,k−1] the road from pipi to pi+1pi+1 exists, so the path goes along the roads of Bertown. OutputPrint two integers: the minimum and the maximum number of rebuilds that could have happened during the journey.ExamplesInputCopy6 9
1 5
5 4
1 2
2 3
3 4
4 1
2 6
6 4
4 2
4
1 2 3 4
OutputCopy1 2
InputCopy7 7
1 2
2 3
3 4
4 5
5 6
6 7
7 1
7
1 2 3 4 5 6 7
OutputCopy0 0
InputCopy8 13
8 7
8 6
7 5
7 4
6 5
6 4
5 3
5 2
4 3
4 2
3 1
2 1
1 8
5
8 7 5 2 1
OutputCopy0 3
| [
"dfs and similar",
"graphs",
"shortest paths"
] | from collections import deque
n,m=map(int,input().split())
h=dict()
for i in range( m):
u,v=map(int,input().split())
u-=1
v-=1
if v in h:
h[v].append(u)
else:
h[v]=[u]
k=int(input())
arr=[int(j)-1 for j in input().split()]
p=deque([arr[-1]])
h1=dict()
h1[arr[-1]]=1
E = [0] * n
while(p):
for nd in h[p[0]]:
if nd not in h1:
p.append(nd)
h1[nd]=h1[p[0]]+1
else:
if h1[p[0]]+1==h1[nd]:
E[nd]+=1
p.popleft()
R = S = 0
for i in range(1, k):
u, v = arr[i - 1], arr[i]
if u not in h1:
h1[u]=-1
if v not in h1:
h1[v]=-1
if h1[u] <= h1[v]:
R += 1
S += 1
elif E[u]:
S += 1
print(R, S)
| py |
1323 | B | B. Count Subrectanglestime limit per test1 secondmemory limit per test512 megabytesinputstandard inputoutputstandard outputYou are given an array aa of length nn and array bb of length mm both consisting of only integers 00 and 11. Consider a matrix cc of size n×mn×m formed by following rule: ci,j=ai⋅bjci,j=ai⋅bj (i.e. aiai multiplied by bjbj). It's easy to see that cc consists of only zeroes and ones too.How many subrectangles of size (area) kk consisting only of ones are there in cc?A subrectangle is an intersection of a consecutive (subsequent) segment of rows and a consecutive (subsequent) segment of columns. I.e. consider four integers x1,x2,y1,y2x1,x2,y1,y2 (1≤x1≤x2≤n1≤x1≤x2≤n, 1≤y1≤y2≤m1≤y1≤y2≤m) a subrectangle c[x1…x2][y1…y2]c[x1…x2][y1…y2] is an intersection of the rows x1,x1+1,x1+2,…,x2x1,x1+1,x1+2,…,x2 and the columns y1,y1+1,y1+2,…,y2y1,y1+1,y1+2,…,y2.The size (area) of a subrectangle is the total number of cells in it.InputThe first line contains three integers nn, mm and kk (1≤n,m≤40000,1≤k≤n⋅m1≤n,m≤40000,1≤k≤n⋅m), length of array aa, length of array bb and required size of subrectangles.The second line contains nn integers a1,a2,…,ana1,a2,…,an (0≤ai≤10≤ai≤1), elements of aa.The third line contains mm integers b1,b2,…,bmb1,b2,…,bm (0≤bi≤10≤bi≤1), elements of bb.OutputOutput single integer — the number of subrectangles of cc with size (area) kk consisting only of ones.ExamplesInputCopy3 3 2
1 0 1
1 1 1
OutputCopy4
InputCopy3 5 4
1 1 1
1 1 1 1 1
OutputCopy14
NoteIn first example matrix cc is: There are 44 subrectangles of size 22 consisting of only ones in it: In second example matrix cc is: | [
"binary search",
"greedy",
"implementation"
] | import sys
import os
from io import BytesIO, IOBase
BUFSIZE = 8192
class FastIO(IOBase):
newlines = 0
def __init__(self, file):
self._fd = file.fileno()
self.buffer = BytesIO()
self.writable = "x" in file.mode or "r" not in file.mode
self.write = self.buffer.write if self.writable else None
def read(self):
while True:
b = os.read(self._fd, max(os.fstat(self._fd).st_size, BUFSIZE))
if not b:
break
ptr = self.buffer.tell()
self.buffer.seek(0, 2), self.buffer.write(b), self.buffer.seek(ptr)
self.newlines = 0
return self.buffer.read()
def readline(self):
while self.newlines == 0:
b = os.read(self._fd, max(os.fstat(self._fd).st_size, BUFSIZE))
self.newlines = b.count(b"\n") + (not b)
ptr = self.buffer.tell()
self.buffer.seek(0, 2), self.buffer.write(b), self.buffer.seek(ptr)
self.newlines -= 1
return self.buffer.readline()
def flush(self):
if self.writable:
os.write(self._fd, self.buffer.getvalue())
self.buffer.truncate(0), self.buffer.seek(0)
class IOWrapper(IOBase):
def __init__(self, file):
self.buffer = FastIO(file)
self.flush = self.buffer.flush
self.writable = self.buffer.writable
self.write = lambda s: self.buffer.write(s.encode("ascii"))
self.read = lambda: self.buffer.read().decode("ascii")
self.readline = lambda: self.buffer.readline().decode("ascii")
sys.stdin, sys.stdout = IOWrapper(sys.stdin), IOWrapper(sys.stdout)
input = lambda: sys.stdin.readline().rstrip("\r\n")
def fctrs(n):
import math
res=[1, n]
for i in range(2, int(math.sqrt(n))+1):
if n%i==0:
res.append(i)
if i!=(n//i):
res.append(n//i)
return res
def cons(arr):
s=0
res=[0]*(1+len(arr))
for i in range(len(arr)):
if arr[i]==1:
s+=1
else:
s=0
res[s]+=1
s=0
for i in range(len(arr), -1, -1):
res[i]+=s
s=res[i]
return res
n, m, k=map(int, input().split())
a=[int(i) for i in input().split()]
b=[int(i) for i in input().split()]
bcons=cons(b)
acons=cons(a)
# print(acons)
# print(bcons)
ans=0
for i in range(1, len(bcons)):
if k%i==0:
c=(k//i)
a1=(acons[c] if c<len(acons) else 0)
b1=bcons[i]
# print(i, c, a1, b1)
ans+=a1*b1
print(ans) | py |
1304 | B | B. Longest Palindrometime limit per test1 secondmemory limit per test256 megabytesinputstandard inputoutputstandard outputReturning back to problem solving; Gildong is now studying about palindromes. He learned that a palindrome is a string that is the same as its reverse. For example, strings "pop", "noon", "x", and "kkkkkk" are palindromes, while strings "moon", "tv", and "abab" are not. An empty string is also a palindrome.Gildong loves this concept so much, so he wants to play with it. He has nn distinct strings of equal length mm. He wants to discard some of the strings (possibly none or all) and reorder the remaining strings so that the concatenation becomes a palindrome. He also wants the palindrome to be as long as possible. Please help him find one.InputThe first line contains two integers nn and mm (1≤n≤1001≤n≤100, 1≤m≤501≤m≤50) — the number of strings and the length of each string.Next nn lines contain a string of length mm each, consisting of lowercase Latin letters only. All strings are distinct.OutputIn the first line, print the length of the longest palindrome string you made.In the second line, print that palindrome. If there are multiple answers, print any one of them. If the palindrome is empty, print an empty line or don't print this line at all.ExamplesInputCopy3 3
tab
one
bat
OutputCopy6
tabbat
InputCopy4 2
oo
ox
xo
xx
OutputCopy6
oxxxxo
InputCopy3 5
hello
codef
orces
OutputCopy0
InputCopy9 4
abab
baba
abcd
bcde
cdef
defg
wxyz
zyxw
ijji
OutputCopy20
ababwxyzijjizyxwbaba
NoteIn the first example; "battab" is also a valid answer.In the second example; there can be 4 different valid answers including the sample output. We are not going to provide any hints for what the others are.In the third example; the empty string is the only valid palindrome string. | [
"brute force",
"constructive algorithms",
"greedy",
"implementation",
"strings"
] | def PROBLEM():
n,m=map(int,input().split())
T=[]
for _ in range(n):
T.append(input())
S=''
i=0
Nb=1
while T!=[]:
P=RecherchePosPali(T,n,T[i])
if P!=-1:
S=T[i]+S+T[P]
T.remove(T[P])
T.remove(T[i])
n-=2
elif Nb and S[:len(S)//2]+T[i]+S[len(S)//2:]==(S[:len(S)//2]+T[i]+S[len(S)//2:])[::-1]:
S=S[:len(S)//2]+T[i]+S[len(S)//2:]
T.remove(T[i])
n-=1
Nb=0
else:
T.remove(T[i])
n-=1
print(len(S))
print(S)
def RecherchePosPali(T,n,S):
P=-1
i=1
while P==-1 and i<n:
if S==T[i][::-1]:
P=i
else:
i+=1
return P
PROBLEM()
| py |
1305 | C | C. Kuroni and Impossible Calculationtime limit per test1 secondmemory limit per test256 megabytesinputstandard inputoutputstandard outputTo become the king of Codeforces; Kuroni has to solve the following problem.He is given nn numbers a1,a2,…,ana1,a2,…,an. Help Kuroni to calculate ∏1≤i<j≤n|ai−aj|∏1≤i<j≤n|ai−aj|. As result can be very big, output it modulo mm.If you are not familiar with short notation, ∏1≤i<j≤n|ai−aj|∏1≤i<j≤n|ai−aj| is equal to |a1−a2|⋅|a1−a3|⋅|a1−a2|⋅|a1−a3|⋅ …… ⋅|a1−an|⋅|a2−a3|⋅|a2−a4|⋅⋅|a1−an|⋅|a2−a3|⋅|a2−a4|⋅ …… ⋅|a2−an|⋅⋅|a2−an|⋅ …… ⋅|an−1−an|⋅|an−1−an|. In other words, this is the product of |ai−aj||ai−aj| for all 1≤i<j≤n1≤i<j≤n.InputThe first line contains two integers nn, mm (2≤n≤2⋅1052≤n≤2⋅105, 1≤m≤10001≤m≤1000) — number of numbers and modulo.The second line contains nn integers a1,a2,…,ana1,a2,…,an (0≤ai≤1090≤ai≤109).OutputOutput the single number — ∏1≤i<j≤n|ai−aj|modm∏1≤i<j≤n|ai−aj|modm.ExamplesInputCopy2 10
8 5
OutputCopy3InputCopy3 12
1 4 5
OutputCopy0InputCopy3 7
1 4 9
OutputCopy1NoteIn the first sample; |8−5|=3≡3mod10|8−5|=3≡3mod10.In the second sample, |1−4|⋅|1−5|⋅|4−5|=3⋅4⋅1=12≡0mod12|1−4|⋅|1−5|⋅|4−5|=3⋅4⋅1=12≡0mod12.In the third sample, |1−4|⋅|1−9|⋅|4−9|=3⋅8⋅5=120≡1mod7|1−4|⋅|1−9|⋅|4−9|=3⋅8⋅5=120≡1mod7. | [
"brute force",
"combinatorics",
"math",
"number theory"
] | from sys import stdin
input = stdin.readline
n, m = [int(x) for x in input().split()]
a = [int(x) for x in input().split()]
if n > m: print(0); quit()
ans = 1
for i in range(n):
for j in range(i):
ans = ans*abs(a[i]-a[j])%m
print(ans)
# https://www.bilibili.com/read/cv8299203 | py |
1295 | A | A. Display The Numbertime limit per test1 secondmemory limit per test256 megabytesinputstandard inputoutputstandard outputYou have a large electronic screen which can display up to 998244353998244353 decimal digits. The digits are displayed in the same way as on different electronic alarm clocks: each place for a digit consists of 77 segments which can be turned on and off to compose different digits. The following picture describes how you can display all 1010 decimal digits:As you can see, different digits may require different number of segments to be turned on. For example, if you want to display 11, you have to turn on 22 segments of the screen, and if you want to display 88, all 77 segments of some place to display a digit should be turned on.You want to display a really large integer on the screen. Unfortunately, the screen is bugged: no more than nn segments can be turned on simultaneously. So now you wonder what is the greatest integer that can be displayed by turning on no more than nn segments.Your program should be able to process tt different test cases.InputThe first line contains one integer tt (1≤t≤1001≤t≤100) — the number of test cases in the input.Then the test cases follow, each of them is represented by a separate line containing one integer nn (2≤n≤1052≤n≤105) — the maximum number of segments that can be turned on in the corresponding testcase.It is guaranteed that the sum of nn over all test cases in the input does not exceed 105105.OutputFor each test case, print the greatest integer that can be displayed by turning on no more than nn segments of the screen. Note that the answer may not fit in the standard 3232-bit or 6464-bit integral data type.ExampleInputCopy2
3
4
OutputCopy7
11
| [
"greedy"
] | for run in range(int(input())):
n=int(input())
s=''
if n%2==1:
s+='7'
n-=3
while n>0:
s+='1'
n-=2
print(s) | py |
1307 | A | A. Cow and Haybalestime limit per test2 secondsmemory limit per test256 megabytesinputstandard inputoutputstandard outputThe USA Construction Operation (USACO) recently ordered Farmer John to arrange a row of nn haybale piles on the farm. The ii-th pile contains aiai haybales. However, Farmer John has just left for vacation, leaving Bessie all on her own. Every day, Bessie the naughty cow can choose to move one haybale in any pile to an adjacent pile. Formally, in one day she can choose any two indices ii and jj (1≤i,j≤n1≤i,j≤n) such that |i−j|=1|i−j|=1 and ai>0ai>0 and apply ai=ai−1ai=ai−1, aj=aj+1aj=aj+1. She may also decide to not do anything on some days because she is lazy.Bessie wants to maximize the number of haybales in pile 11 (i.e. to maximize a1a1), and she only has dd days to do so before Farmer John returns. Help her find the maximum number of haybales that may be in pile 11 if she acts optimally!InputThe input consists of multiple test cases. The first line contains an integer tt (1≤t≤1001≤t≤100) — the number of test cases. Next 2t2t lines contain a description of test cases — two lines per test case.The first line of each test case contains integers nn and dd (1≤n,d≤1001≤n,d≤100) — the number of haybale piles and the number of days, respectively. The second line of each test case contains nn integers a1,a2,…,ana1,a2,…,an (0≤ai≤1000≤ai≤100) — the number of haybales in each pile.OutputFor each test case, output one integer: the maximum number of haybales that may be in pile 11 after dd days if Bessie acts optimally.ExampleInputCopy3
4 5
1 0 3 2
2 2
100 1
1 8
0
OutputCopy3
101
0
NoteIn the first test case of the sample; this is one possible way Bessie can end up with 33 haybales in pile 11: On day one, move a haybale from pile 33 to pile 22 On day two, move a haybale from pile 33 to pile 22 On day three, move a haybale from pile 22 to pile 11 On day four, move a haybale from pile 22 to pile 11 On day five, do nothing In the second test case of the sample, Bessie can do nothing on the first day and move a haybale from pile 22 to pile 11 on the second day. | [
"greedy",
"implementation"
] | from collections import *
from heapq import *
from bisect import *
from itertools import *
from functools import *
from math import *
from string import *
import operator
import sys
input = sys.stdin.readline
def solve():
n, d = map(int, input().split())
A = list(map(int, input().split()))
cost = lambda i: i
ans = A[0]
for i in range(1, n):
moved = min(d // cost(i), A[i])
ans += moved
d -= moved * cost(i)
print(ans)
def main():
tests = int(input())
for _ in range(tests):
solve()
if __name__ == "__main__":
main()
| py |
1141 | G | G. Privatization of Roads in Treelandtime limit per test2 secondsmemory limit per test256 megabytesinputstandard inputoutputstandard outputTreeland consists of nn cities and n−1n−1 roads. Each road is bidirectional and connects two distinct cities. From any city you can get to any other city by roads. Yes, you are right — the country's topology is an undirected tree.There are some private road companies in Treeland. The government decided to sell roads to the companies. Each road will belong to one company and a company can own multiple roads.The government is afraid to look unfair. They think that people in a city can consider them unfair if there is one company which owns two or more roads entering the city. The government wants to make such privatization that the number of such cities doesn't exceed kk and the number of companies taking part in the privatization is minimal.Choose the number of companies rr such that it is possible to assign each road to one company in such a way that the number of cities that have two or more roads of one company is at most kk. In other words, if for a city all the roads belong to the different companies then the city is good. Your task is to find the minimal rr that there is such assignment to companies from 11 to rr that the number of cities which are not good doesn't exceed kk. The picture illustrates the first example (n=6,k=2n=6,k=2). The answer contains r=2r=2 companies. Numbers on the edges denote edge indices. Edge colors mean companies: red corresponds to the first company, blue corresponds to the second company. The gray vertex (number 33) is not good. The number of such vertices (just one) doesn't exceed k=2k=2. It is impossible to have at most k=2k=2 not good cities in case of one company. InputThe first line contains two integers nn and kk (2≤n≤200000,0≤k≤n−12≤n≤200000,0≤k≤n−1) — the number of cities and the maximal number of cities which can have two or more roads belonging to one company.The following n−1n−1 lines contain roads, one road per line. Each line contains a pair of integers xixi, yiyi (1≤xi,yi≤n1≤xi,yi≤n), where xixi, yiyi are cities connected with the ii-th road.OutputIn the first line print the required rr (1≤r≤n−11≤r≤n−1). In the second line print n−1n−1 numbers c1,c2,…,cn−1c1,c2,…,cn−1 (1≤ci≤r1≤ci≤r), where cici is the company to own the ii-th road. If there are multiple answers, print any of them.ExamplesInputCopy6 2
1 4
4 3
3 5
3 6
5 2
OutputCopy2
1 2 1 1 2 InputCopy4 2
3 1
1 4
1 2
OutputCopy1
1 1 1 InputCopy10 2
10 3
1 2
1 3
1 4
2 5
2 6
2 7
3 8
3 9
OutputCopy3
1 1 2 3 2 3 1 3 1 | [
"binary search",
"constructive algorithms",
"dfs and similar",
"graphs",
"greedy",
"trees"
] | def strelizia():
oh_pie = 0
kaguya = 0
owe_pie = [(0, 0, 0)]
while (kaguya <= oh_pie):
o_pie = oh_pie
while (kaguya <= o_pie):
x = owe_pie[kaguya][0]
p = owe_pie[kaguya][1]
c = 1
l = len(stamen[x])
if (l > virm):
for (to, ed) in stamen[x]:
if (to != p):
darling[ed - 1] = c
owe_pie.append((to, x, c))
oh_pie += 1
else:
for (to, ed) in stamen[x]:
if (c == owe_pie[kaguya][2]):
c += 1
if (to != p):
darling[ed - 1] = c
owe_pie.append((to, x, c))
oh_pie += 1
c += 1
kaguya += 1
darling = []
franxx = input().split()
pistil = []
stamen = []
for i in range(0, int(franxx[0])):
pistil.append(0)
stamen.append([])
for i in range(1, int(franxx[0])):
darling.append(0)
edge = input().split()
stamen[int(edge[0]) - 1].append((int(edge[1]) - 1, i))
stamen[int(edge[1]) - 1].append((int(edge[0]) - 1, i))
pistil[int(edge[0]) - 1] += 1
pistil[int(edge[1]) - 1] += 1
pistil.sort()
virm = pistil[int(franxx[0]) - int(franxx[1]) - 1]
print(virm)
strelizia()
for i in range(1, int(franxx[0])):
print(darling[i - 1], end = " ") | py |
1311 | F | F. Moving Pointstime limit per test2 secondsmemory limit per test256 megabytesinputstandard inputoutputstandard outputThere are nn points on a coordinate axis OXOX. The ii-th point is located at the integer point xixi and has a speed vivi. It is guaranteed that no two points occupy the same coordinate. All nn points move with the constant speed, the coordinate of the ii-th point at the moment tt (tt can be non-integer) is calculated as xi+t⋅vixi+t⋅vi.Consider two points ii and jj. Let d(i,j)d(i,j) be the minimum possible distance between these two points over any possible moments of time (even non-integer). It means that if two points ii and jj coincide at some moment, the value d(i,j)d(i,j) will be 00.Your task is to calculate the value ∑1≤i<j≤n∑1≤i<j≤n d(i,j)d(i,j) (the sum of minimum distances over all pairs of points).InputThe first line of the input contains one integer nn (2≤n≤2⋅1052≤n≤2⋅105) — the number of points.The second line of the input contains nn integers x1,x2,…,xnx1,x2,…,xn (1≤xi≤1081≤xi≤108), where xixi is the initial coordinate of the ii-th point. It is guaranteed that all xixi are distinct.The third line of the input contains nn integers v1,v2,…,vnv1,v2,…,vn (−108≤vi≤108−108≤vi≤108), where vivi is the speed of the ii-th point.OutputPrint one integer — the value ∑1≤i<j≤n∑1≤i<j≤n d(i,j)d(i,j) (the sum of minimum distances over all pairs of points).ExamplesInputCopy3
1 3 2
-100 2 3
OutputCopy3
InputCopy5
2 1 4 3 5
2 2 2 3 4
OutputCopy19
InputCopy2
2 1
-3 0
OutputCopy0
| [
"data structures",
"divide and conquer",
"implementation",
"sortings"
] | import sys
def main():
input = sys.stdin.readline
_ = int(input())
*X, = map(int, input().split())
*V, = map(int, input().split())
D = {v: i for i, v in enumerate(sorted(set(V)))}
N = len(D)
ans = 0
I = sorted(range(len(X)), key=lambda i: X[i])
C = BinaryIndexedTree(N)
S = BinaryIndexedTree(N)
for i in I:
x, v = X[i], V[i]
j = D[v]
c = C.query(j)
s = S.query(j)
ans += x * c - s
C.add(j, 1)
S.add(j, x)
print(ans)
class BinaryIndexedTree:
__slots__ = ('__n', '__d', '__f', '__id')
def __init__(self, n=None, f=lambda x, y: x + y, identity=0, initial_values=None):
assert(n or initial_values)
self.__f, self.__id, = f, identity
self.__n = len(initial_values) if initial_values else n
self.__d = [identity] * (self.__n + 1)
if initial_values:
for i, v in enumerate(initial_values): self.add(i, v)
def add(self, i, v):
n, f, d = self.__n, self.__f, self.__d
i += 1
while i <= n:
d[i] = f(d[i], v)
i += -i & i
def query(self, r):
res, f, d = self.__id, self.__f, self.__d
r += 1
while r:
res = f(res, d[r])
r -= -r & r
return res
if __name__ == '__main__':
main()
| py |
1294 | C | C. Product of Three Numberstime limit per test2 secondsmemory limit per test256 megabytesinputstandard inputoutputstandard outputYou are given one integer number nn. Find three distinct integers a,b,ca,b,c such that 2≤a,b,c2≤a,b,c and a⋅b⋅c=na⋅b⋅c=n or say that it is impossible to do it.If there are several answers, you can print any.You have to answer tt independent test cases.InputThe first line of the input contains one integer tt (1≤t≤1001≤t≤100) — the number of test cases.The next nn lines describe test cases. The ii-th test case is given on a new line as one integer nn (2≤n≤1092≤n≤109).OutputFor each test case, print the answer on it. Print "NO" if it is impossible to represent nn as a⋅b⋅ca⋅b⋅c for some distinct integers a,b,ca,b,c such that 2≤a,b,c2≤a,b,c.Otherwise, print "YES" and any possible such representation.ExampleInputCopy5
64
32
97
2
12345
OutputCopyYES
2 4 8
NO
NO
NO
YES
3 5 823
| [
"greedy",
"math",
"number theory"
] | from sys import stdin
input=lambda :stdin.readline()[:-1]
def solve():
n=int(input())
ans=[]
now=n
for i in range(2,int(n**0.5)+10):
if now%i==0 and len(ans)<=1:
ans.append(i)
now//=i
ans.append(now)
if len(set(ans))==3 and min(ans)!=1:
print('YES')
print(*ans)
else:
print('NO')
for _ in range(int(input())):
solve() | py |
1316 | A | A. Grade Allocationtime limit per test1 secondmemory limit per test256 megabytesinputstandard inputoutputstandard outputnn students are taking an exam. The highest possible score at this exam is mm. Let aiai be the score of the ii-th student. You have access to the school database which stores the results of all students.You can change each student's score as long as the following conditions are satisfied: All scores are integers 0≤ai≤m0≤ai≤m The average score of the class doesn't change. You are student 11 and you would like to maximize your own score.Find the highest possible score you can assign to yourself such that all conditions are satisfied.InputEach test contains multiple test cases. The first line contains the number of test cases tt (1≤t≤2001≤t≤200). The description of the test cases follows.The first line of each test case contains two integers nn and mm (1≤n≤1031≤n≤103, 1≤m≤1051≤m≤105) — the number of students and the highest possible score respectively.The second line of each testcase contains nn integers a1,a2,…,ana1,a2,…,an (0≤ai≤m0≤ai≤m) — scores of the students.OutputFor each testcase, output one integer — the highest possible score you can assign to yourself such that both conditions are satisfied._ExampleInputCopy2
4 10
1 2 3 4
4 5
1 2 3 4
OutputCopy10
5
NoteIn the first case; a=[1,2,3,4]a=[1,2,3,4], with average of 2.52.5. You can change array aa to [10,0,0,0][10,0,0,0]. Average remains 2.52.5, and all conditions are satisfied.In the second case, 0≤ai≤50≤ai≤5. You can change aa to [5,1,1,3][5,1,1,3]. You cannot increase a1a1 further as it will violate condition 0≤ai≤m0≤ai≤m. | [
"implementation"
] | q = int(input())
for i in range(q):
w = input().split()
n, m = w
n = int(n)
m = int(m)
b = input().split()
s = 0
for j in range(len(b)):
b[j] = int(b[j])
s = s + b[j]
if s > m:
print(m)
else:
print(s)
| py |
1141 | A | A. Game 23time limit per test1 secondmemory limit per test256 megabytesinputstandard inputoutputstandard outputPolycarp plays "Game 23". Initially he has a number nn and his goal is to transform it to mm. In one move, he can multiply nn by 22 or multiply nn by 33. He can perform any number of moves.Print the number of moves needed to transform nn to mm. Print -1 if it is impossible to do so.It is easy to prove that any way to transform nn to mm contains the same number of moves (i.e. number of moves doesn't depend on the way of transformation).InputThe only line of the input contains two integers nn and mm (1≤n≤m≤5⋅1081≤n≤m≤5⋅108).OutputPrint the number of moves to transform nn to mm, or -1 if there is no solution.ExamplesInputCopy120 51840
OutputCopy7
InputCopy42 42
OutputCopy0
InputCopy48 72
OutputCopy-1
NoteIn the first example; the possible sequence of moves is: 120→240→720→1440→4320→12960→25920→51840.120→240→720→1440→4320→12960→25920→51840. The are 77 steps in total.In the second example, no moves are needed. Thus, the answer is 00.In the third example, it is impossible to transform 4848 to 7272. | [
"implementation",
"math"
] | a = input().split()
b = int(a[1])
a = int(a[0])
if a == b: print("0")
elif b % a != 0 or (b / a % 2 != 0 and b / a % 3 != 0): print("-1")
else:
b /= a
count = 0
while b % 2 == 0:
b /= 2
count += 1
while b % 3 == 0:
b /= 3
count += 1
if b != 1: print("-1")
else: print(count) | py |
1303 | B | B. National Projecttime limit per test2 secondsmemory limit per test256 megabytesinputstandard inputoutputstandard outputYour company was appointed to lay new asphalt on the highway of length nn. You know that every day you can either repair one unit of the highway (lay new asphalt over one unit of the highway) or skip repairing.Skipping the repair is necessary because of the climate. The climate in your region is periodical: there are gg days when the weather is good and if you lay new asphalt these days it becomes high-quality pavement; after that, the weather during the next bb days is bad, and if you lay new asphalt these days it becomes low-quality pavement; again gg good days, bb bad days and so on.You can be sure that you start repairing at the start of a good season, in other words, days 1,2,…,g1,2,…,g are good.You don't really care about the quality of the highway, you just want to make sure that at least half of the highway will have high-quality pavement. For example, if the n=5n=5 then at least 33 units of the highway should have high quality; if n=4n=4 then at least 22 units should have high quality.What is the minimum number of days is needed to finish the repair of the whole highway?InputThe first line contains a single integer TT (1≤T≤1041≤T≤104) — the number of test cases.Next TT lines contain test cases — one per line. Each line contains three integers nn, gg and bb (1≤n,g,b≤1091≤n,g,b≤109) — the length of the highway and the number of good and bad days respectively.OutputPrint TT integers — one per test case. For each test case, print the minimum number of days required to repair the whole highway if at least half of it should have high quality.ExampleInputCopy3
5 1 1
8 10 10
1000000 1 1000000
OutputCopy5
8
499999500000
NoteIn the first test case; you can just lay new asphalt each day; since days 1,3,51,3,5 are good.In the second test case, you can also lay new asphalt each day, since days 11-88 are good. | [
"math"
] | def solve():
n, g, b = read_ints()
good_needed = 0--n // 2
loops = 0--good_needed // g
cnt = (loops - 1) * (g + b)
diff = good_needed - (loops - 1) * g
cnt += diff
if cnt < n:
cnt = n
return cnt
def main():
t = int(input())
output = []
for _ in range(t):
ans = solve()
output.append(ans)
print_lines(output)
def input(): return next(test).strip()
def read_ints(): return [int(c) for c in input().split()]
def print_lines(lst): print('\n'.join(map(str, lst)))
if __name__ == "__main__":
import sys
from os import environ as env
if 'COMPUTERNAME' in env and 'L2A6HRI' in env['COMPUTERNAME']:
sys.stdout = open('out.txt', 'w')
sys.stdin = open('in.txt', 'r')
test = iter(sys.stdin.readlines())
main()
| py |
1305 | C | C. Kuroni and Impossible Calculationtime limit per test1 secondmemory limit per test256 megabytesinputstandard inputoutputstandard outputTo become the king of Codeforces; Kuroni has to solve the following problem.He is given nn numbers a1,a2,…,ana1,a2,…,an. Help Kuroni to calculate ∏1≤i<j≤n|ai−aj|∏1≤i<j≤n|ai−aj|. As result can be very big, output it modulo mm.If you are not familiar with short notation, ∏1≤i<j≤n|ai−aj|∏1≤i<j≤n|ai−aj| is equal to |a1−a2|⋅|a1−a3|⋅|a1−a2|⋅|a1−a3|⋅ …… ⋅|a1−an|⋅|a2−a3|⋅|a2−a4|⋅⋅|a1−an|⋅|a2−a3|⋅|a2−a4|⋅ …… ⋅|a2−an|⋅⋅|a2−an|⋅ …… ⋅|an−1−an|⋅|an−1−an|. In other words, this is the product of |ai−aj||ai−aj| for all 1≤i<j≤n1≤i<j≤n.InputThe first line contains two integers nn, mm (2≤n≤2⋅1052≤n≤2⋅105, 1≤m≤10001≤m≤1000) — number of numbers and modulo.The second line contains nn integers a1,a2,…,ana1,a2,…,an (0≤ai≤1090≤ai≤109).OutputOutput the single number — ∏1≤i<j≤n|ai−aj|modm∏1≤i<j≤n|ai−aj|modm.ExamplesInputCopy2 10
8 5
OutputCopy3InputCopy3 12
1 4 5
OutputCopy0InputCopy3 7
1 4 9
OutputCopy1NoteIn the first sample; |8−5|=3≡3mod10|8−5|=3≡3mod10.In the second sample, |1−4|⋅|1−5|⋅|4−5|=3⋅4⋅1=12≡0mod12|1−4|⋅|1−5|⋅|4−5|=3⋅4⋅1=12≡0mod12.In the third sample, |1−4|⋅|1−9|⋅|4−9|=3⋅8⋅5=120≡1mod7|1−4|⋅|1−9|⋅|4−9|=3⋅8⋅5=120≡1mod7. | [
"brute force",
"combinatorics",
"math",
"number theory"
] | ######################################################################################
#------------------------------------Template---------------------------------------#
######################################################################################
import collections
import heapq
import sys
import math
import itertools
import bisect
from io import BytesIO, IOBase
import os
def vsInput():
sys.stdin = open('input.txt', 'r')
sys.stdout = open('output.txt', 'w')
BUFSIZE = 8192
class FastIO(IOBase):
newlines = 0
def __init__(self, file):
self._fd = file.fileno()
self.buffer = BytesIO()
self.writable = "x" in file.mode or "r" not in file.mode
self.write = self.buffer.write if self.writable else None
def read(self):
while True:
b = os.read(self._fd, max(os.fstat(self._fd).st_size, BUFSIZE))
if not b:
break
ptr = self.buffer.tell()
self.buffer.seek(0, 2), self.buffer.write(b), self.buffer.seek(ptr)
self.newlines = 0
return self.buffer.read()
def readline(self):
while self.newlines == 0:
b = os.read(self._fd, max(os.fstat(self._fd).st_size, BUFSIZE))
self.newlines = b.count(b"\n") + (not b)
ptr = self.buffer.tell()
self.buffer.seek(0, 2), self.buffer.write(b), self.buffer.seek(ptr)
self.newlines -= 1
return self.buffer.readline()
def flush(self):
if self.writable:
os.write(self._fd, self.buffer.getvalue())
self.buffer.truncate(0), self.buffer.seek(0)
class IOWrapper(IOBase):
def __init__(self, file):
self.buffer = FastIO(file)
self.flush = self.buffer.flush
self.writable = self.buffer.writable
self.write = lambda s: self.buffer.write(s.encode("ascii"))
self.read = lambda: self.buffer.read().decode("ascii")
self.readline = lambda: self.buffer.readline().decode("ascii")
sys.stdin, sys.stdout = IOWrapper(sys.stdin), IOWrapper(sys.stdout)
def input(): return sys.stdin.readline().rstrip("\r\n")
######################################################################################
#--------------------------------------Input-----------------------------------------#
######################################################################################
def value(): return tuple(map(int, input().split()))
def inlt(): return [int(i) for i in input().split()]
def inp(): return int(input())
def insr(): return input()
def stlt(): return [i for i in input().split()]
######################################################################################
#------------------------------------Functions---------------------------------------#
######################################################################################
def SieveOfEratosthenes(n):
prime = [True for i in range(n+1)]
p = 2
l = []
while (p * p <= n):
if (prime[p] == True):
for i in range(p * p, n+1, p):
prime[i] = False
p += 1
for p in range(2, n+1):
if prime[p]:
l.append(p)
# print(p)
else:
continue
return l
def isPrime(n):
prime_flag = 0
if n > 1:
for i in range(2, int(math.sqrt(n)) + 1):
if n % i == 0:
prime_flag = 1
break
if prime_flag == 0:
return True
else:
return False
else:
return False
def gcdofarray(a):
x = 0
for p in a:
x = math.gcd(x, p)
return x
def printDivisors(n):
i = 1
ans = []
while i <= math.sqrt(n):
if (n % i == 0):
if (n / i == i):
ans.append(i)
else:
ans.append(i)
ans.append(n // i)
i = i + 1
ans.sort()
return ans
def binaryToDecimal(n):
return int(n, 2)
def countTriplets(a, n):
s = set()
for i in range(n):
s.add(a[i])
count = 0
for i in range(n):
for j in range(i + 1, n, 1):
xr = a[i] ^ a[j]
if xr in s and xr != a[i] and xr != a[j]:
count += 1
return int(count // 3)
def countOdd(L, R):
N = (R - L) // 2
if (R % 2 != 0 or L % 2 != 0):
N += 1
return N
def isPalindrome(s):
return s == s[::-1]
def sufsum(a):
test_list = a.copy()
test_list.reverse()
n = len(test_list)
for i in range(1, n):
test_list[i] = test_list[i] + test_list[i - 1]
return test_list
def prsum(b):
a = b.copy()
for i in range(1, len(a)):
a[i] += a[i - 1]
return a
def badachotabadachota(nums):
nums.sort()
i = 0
j = len(nums) - 1
ans = []
cc = 0
while len(ans) != len(nums):
if cc % 2 == 0:
ans.append(nums[j])
j -= 1
else:
ans.append(nums[i])
i += 1
cc += 1
return ans
def chotabadachotabada(nums):
nums.sort()
i = 0
j = len(nums) - 1
ans = []
cc = 0
while len(ans) != len(nums):
if cc % 2 == 1:
ans.append(nums[j])
j -= 1
else:
ans.append(nums[i])
i += 1
cc += 1
return ans
def primeFactors(n):
ans = []
while n % 2 == 0:
ans.append(2)
n = n // 2
for i in range(3, int(math.sqrt(n))+1, 2):
while n % i == 0:
ans.append(i)
n = n / i
if n > 2:
ans.append(n)
return ans
def closestMultiple(n, x):
if x > n:
return x
z = (int)(x / 2)
n = n + z
n = n - (n % x)
return n
def getPairsCount(arr, n, sum):
m = [0] * 1000
for i in range(0, n):
m[arr[i]] += 1
twice_count = 0
for i in range(0, n):
twice_count += m[int(sum - arr[i])]
if (int(sum - arr[i]) == arr[i]):
twice_count -= 1
return int(twice_count / 2)
def remove_consec_duplicates(test_list):
res = [i[0] for i in itertools.groupby(test_list)]
return res
def BigPower(a, b, mod):
if b == 0:
return 1
ans = BigPower(a, b//2, mod)
ans *= ans
ans %= mod
if b % 2:
ans *= a
return ans % mod
def nextPowerOf2(n):
count = 0
if (n and not(n & (n - 1))):
return n
while(n != 0):
n >>= 1
count += 1
return 1 << count
# This function multiplies x with the number
# represented by res[]. res_size is size of res[]
# or number of digits in the number represented
# by res[]. This function uses simple school
# mathematics for multiplication. This function
# may value of res_size and returns the new value
# of res_size
def multiply(x, res, res_size):
carry = 0 # Initialize carry
# One by one multiply n with individual
# digits of res[]
i = 0
while i < res_size:
prod = res[i] * x + carry
res[i] = prod % 10 # Store last digit of
# 'prod' in res[]
# make sure floor division is used
carry = prod//10 # Put rest in carry
i = i + 1
# Put carry in res and increase result size
while (carry):
res[res_size] = carry % 10
# make sure floor division is used
# to avoid floating value
carry = carry // 10
res_size = res_size + 1
return res_size
def Kadane(a, size):
max_so_far = -1e18 - 1
max_ending_here = 0
for i in range(0, size):
max_ending_here = max_ending_here + a[i]
if (max_so_far < max_ending_here):
max_so_far = max_ending_here
if max_ending_here < 0:
max_ending_here = 0
return max_so_far
def highestPowerof2(n):
res = 0
for i in range(n, 0, -1):
if ((i & (i - 1)) == 0):
res = i
break
return res
def lower_bound(numbers, length, searchnum):
answer = -1
start = 0
end = length - 1
while start <= end:
middle = (start + end)//2
if numbers[middle] == searchnum:
answer = middle
end = middle - 1
elif numbers[middle] > searchnum:
end = middle - 1
else:
start = middle + 1
return answer
def MEX(nList, start):
nList = set(nList)
mex = start
while mex in nList:
mex += 1
return mex
def MinValue(N, X):
N = list(N)
ln = len(N)
position = ln + 1
# # If the given string N represent
# # a negative value
# if (N[0] == '-'):
# # X must be place at the last
# # index where is greater than N[i]
# for i in range(ln - 1, 0, -1):
# if ((ord(N[i]) - ord('0')) < X):
# position = i
# else:
# # For positive numbers, X must be
# # placed at the last index where
# # it is smaller than N[i]
for i in range(ln - 1, -1, -1):
if ((ord(N[i]) - ord('0')) > X):
position = i
# Insert X at that position
c = chr(X + ord('0'))
str = N.insert(position, c)
# Return the string
return ''.join(N)
def findClosest(arr, n, target):
if (target <= arr[0]):
return arr[0]
if (target >= arr[n - 1]):
return arr[n - 1]
i = 0
j = n
mid = 0
while (i < j):
mid = (i + j) // 2
if (arr[mid] == target):
return arr[mid]
if (target < arr[mid]):
if (mid > 0 and target > arr[mid - 1]):
return getClosest(arr[mid - 1], arr[mid], target)
j = mid
else:
if (mid < n - 1 and target < arr[mid + 1]):
return getClosest(arr[mid], arr[mid + 1], target)
i = mid + 1
return arr[mid]
def getClosest(val1, val2, target):
if (target - val1 >= val2 - target):
return val2
else:
return val1
def ncr(n, r, p):
num = den = 1
for i in range(r):
num = (num * (n - i)) % p
den = (den * (i + 1)) % p
return (num * pow(den, p - 2, p)) % p
# #####################################################################################################
# #------------------------------------------GRAPHS---------------------------------------------------#
# #####################################################################################################
class Graph:
def __init__(self, edges, n):
self.adjList = [[] for _ in range(n)]
for (src, dest) in edges:
self.adjList[src].append(dest)
self.adjList[dest].append(src)
def BFS(graph, v, discovered):
q = collections.deque()
discovered[v] = True
q.append(v)
ans = []
while q:
v = q.popleft()
ans.append(v)
# print(v, end=' ')
for u in graph.adjList[v]:
if not discovered[u]:
discovered[u] = True
q.append(u)
return ans
#############################################################################################################
#--------------------------------------------Fenwick Tree---------------------------------------------------#
#############################################################################################################
def getsumFT(BITTree, i):
s = 0 # initialize result
# index in BITree[] is 1 more than the index in arr[]
i = i+1
# Traverse ancestors of BITree[index]
while i > 0:
# Add current element of BITree to sum
s += BITTree[i]
# Move index to parent node in getSum View
i -= i & (-i)
return s
# Updates a node in Binary Index Tree (BITree) at given index
# in BITree. The given value 'val' is added to BITree[i] and
# all of its ancestors in tree.
def updatebit(BITTree, n, i, v):
# index in BITree[] is 1 more than the index in arr[]
i += 1
# Traverse all ancestors and add 'val'
while i <= n:
# Add 'val' to current node of BI Tree
BITTree[i] += v
# Update index to that of parent in update View
i += i & (-i)
# Constructs and returns a Binary Indexed Tree for given
# array of size n.
def construct(arr, n):
# Create and initialize BITree[] as 0
BITTree = [0]*(n+1)
# Store the actual values in BITree[] using update()
for i in range(n):
updatebit(BITTree, n, i, arr[i])
# Uncomment below lines to see contents of BITree[]
# for i in range(1,n+1):
# print BITTree[i],
return BITTree
#############################################################################################################
#--------------------------------------------Segment Tree---------------------------------------------------#
#############################################################################################################
def getMid(s, e):
return s + (e - s) // 2
""" A recursive function to get the sum of values
in the given range of the array. The following
are parameters for this function.
st --> Pointer to segment tree
si --> Index of current node in the segment tree.
Initially 0 is passed as root is always at index 0
ss & se --> Starting and ending indexes of the segment
represented by current node, i.e., st[si]
qs & qe --> Starting and ending indexes of query range """
def getSumUtil(st, ss, se, qs, qe, si):
# If segment of this node is a part of given range,
# then return the sum of the segment
if (qs <= ss and qe >= se):
return st[si]
# If segment of this node is
# outside the given range
if (se < qs or ss > qe):
return 0
# If a part of this segment overlaps
# with the given range
mid = getMid(ss, se)
return getSumUtil(st, ss, mid, qs, qe, 2 * si + 1) + getSumUtil(st, mid + 1, se, qs, qe, 2 * si + 2)
def updateValueUtil(st, ss, se, i, diff, si):
# Base Case: If the input index lies
# outside the range of this segment
if (i < ss or i > se):
return
# If the input index is in range of this node,
# then update the value of the node and its children
st[si] = st[si] + diff
if (se != ss):
mid = getMid(ss, se)
updateValueUtil(st, ss, mid, i,
diff, 2 * si + 1)
updateValueUtil(st, mid + 1, se, i,
diff, 2 * si + 2)
# The function to update a value in input array
# and segment tree. It uses updateValueUtil()
# to update the value in segment tree
def updateValue(arr, st, n, i, new_val):
# Check for erroneous input index
if (i < 0 or i > n - 1):
print("Invalid Input", end="")
return
# Get the difference between
# new value and old value
diff = new_val - arr[i]
# Update the value in array
arr[i] = new_val
# Update the values of nodes in segment tree
updateValueUtil(st, 0, n - 1, i, diff, 0)
# Return sum of elements in range from
# index qs (query start) to qe (query end).
# It mainly uses getSumUtil()
def getSum(st, n, qs, qe):
# Check for erroneous input values
if (qs < 0 or qe > n - 1 or qs > qe):
print("Invalid Input", end="")
return -1
return getSumUtil(st, 0, n - 1, qs, qe, 0)
# A recursive function that constructs
# Segment Tree for array[ss..se].
# si is index of current node in segment tree st
def constructSTUtil(arr, ss, se, st, si):
# If there is one element in array,
# store it in current node of
# segment tree and return
if (ss == se):
st[si] = arr[ss]
return arr[ss]
# If there are more than one elements,
# then recur for left and right subtrees
# and store the sum of values in this node
mid = getMid(ss, se)
st[si] = constructSTUtil(arr, ss, mid, st, si * 2 + 1) + \
constructSTUtil(arr, mid + 1, se, st, si * 2 + 2)
return st[si]
""" Function to construct segment tree
from given array. This function allocates memory
for segment tree and calls constructSTUtil() to
fill the allocated memory """
def constructST(arr, n):
# Allocate memory for the segment tree
# Height of segment tree
x = (int)(math.ceil(math.log2(n)))
# Maximum size of segment tree
max_size = 2 * (int)(2**x) - 1
# Allocate memory
st = [0] * max_size
# Fill the allocated memory st
constructSTUtil(arr, 0, n - 1, st, 0)
# Return the constructed segment tree
return st
def tobinary(a): return bin(a).replace('0b', '')
def nextPerfectSquare(N):
nextN = math.floor(math.sqrt(N)) + 1
return nextN * nextN
def modFact(n, p):
result = 1
for i in range(1, n + 1):
result *= i
result %= p
return result
def maxsubarrayproduct(arr):
n = len(arr)
max_ending_here = 1
min_ending_here = 1
max_so_far = 0
flag = 0
for i in range(0, n):
if arr[i] > 0:
max_ending_here = max_ending_here * arr[i]
min_ending_here = min (min_ending_here * arr[i], 1)
flag = 1
elif arr[i] == 0:
max_ending_here = 1
min_ending_here = 1
else:
temp = max_ending_here
max_ending_here = max (min_ending_here * arr[i], 1)
min_ending_here = temp * arr[i]
if (max_so_far < max_ending_here):
max_so_far = max_ending_here
if flag == 0 and max_so_far == 0:
return 0
return max_so_far
# #####################################################################################################
alphabets = list("abcdefghijklmnopqrstuvwxyz")
vowels = list("aeiou")
MOD1 = int(1e9 + 7)
MOD2 = 998244353
INF = int(1e18)
# ########################################################################
# #-----------------------------Code Here--------------------------------#
# ########################################################################
# vsInput()
# for _ in range(inp()):
n, m = inlt()
a = inlt()
ans = 0
if n > m:
print(ans)
else:
ans = 1
for i in range(n - 1):
for j in range(i + 1, n):
ans *= abs(a[i] - a[j])
ans %= m
print(ans) | py |
1294 | F | F. Three Paths on a Treetime limit per test2 secondsmemory limit per test256 megabytesinputstandard inputoutputstandard outputYou are given an unweighted tree with nn vertices. Recall that a tree is a connected undirected graph without cycles.Your task is to choose three distinct vertices a,b,ca,b,c on this tree such that the number of edges which belong to at least one of the simple paths between aa and bb, bb and cc, or aa and cc is the maximum possible. See the notes section for a better understanding.The simple path is the path that visits each vertex at most once.InputThe first line contains one integer number nn (3≤n≤2⋅1053≤n≤2⋅105) — the number of vertices in the tree. Next n−1n−1 lines describe the edges of the tree in form ai,biai,bi (1≤ai1≤ai, bi≤nbi≤n, ai≠biai≠bi). It is guaranteed that given graph is a tree.OutputIn the first line print one integer resres — the maximum number of edges which belong to at least one of the simple paths between aa and bb, bb and cc, or aa and cc.In the second line print three integers a,b,ca,b,c such that 1≤a,b,c≤n1≤a,b,c≤n and a≠,b≠c,a≠ca≠,b≠c,a≠c.If there are several answers, you can print any.ExampleInputCopy8
1 2
2 3
3 4
4 5
4 6
3 7
3 8
OutputCopy5
1 8 6
NoteThe picture corresponding to the first example (and another one correct answer):If you choose vertices 1,5,61,5,6 then the path between 11 and 55 consists of edges (1,2),(2,3),(3,4),(4,5)(1,2),(2,3),(3,4),(4,5), the path between 11 and 66 consists of edges (1,2),(2,3),(3,4),(4,6)(1,2),(2,3),(3,4),(4,6) and the path between 55 and 66 consists of edges (4,5),(4,6)(4,5),(4,6). The union of these paths is (1,2),(2,3),(3,4),(4,5),(4,6)(1,2),(2,3),(3,4),(4,5),(4,6) so the answer is 55. It can be shown that there is no better answer. | [
"dfs and similar",
"dp",
"greedy",
"trees"
] | import os, sys
from io import BytesIO, IOBase
from collections import *
class FastIO(IOBase):
newlines = 0
def __init__(self, file):
self._fd = file.fileno()
self.buffer = BytesIO()
self.writable = "x" in file.mode or "r" not in file.mode
self.write = self.buffer.write if self.writable else None
def read(self):
while True:
b = os.read(self._fd, max(os.fstat(self._fd).st_size, 8192))
if not b:
break
ptr = self.buffer.tell()
self.buffer.seek(0, 2), self.buffer.write(b), self.buffer.seek(ptr)
self.newlines = 0
return self.buffer.read()
def readline(self):
while self.newlines == 0:
b = os.read(self._fd, max(os.fstat(self._fd).st_size, 8192))
self.newlines = b.count(b"\n") + (not b)
ptr = self.buffer.tell()
self.buffer.seek(0, 2), self.buffer.write(b), self.buffer.seek(ptr)
self.newlines -= 1
return self.buffer.readline()
def flush(self):
if self.writable:
os.write(self._fd, self.buffer.getvalue())
self.buffer.truncate(0), self.buffer.seek(0)
class IOWrapper(IOBase):
def __init__(self, file):
self.buffer = FastIO(file)
self.flush = self.buffer.flush
self.writable = self.buffer.writable
self.write = lambda s: self.buffer.write(s.encode("ascii"))
self.read = lambda: self.buffer.read().decode("ascii")
self.readline = lambda: self.buffer.readline().decode("ascii")
class graph:
def __init__(self, n):
self.n, self.gdict = n, [[] for _ in range(n + 1)]
def addEdge(self, node1, node2):
self.gdict[node1].append(node2)
self.gdict[node2].append(node1)
def subtree(self, v):
queue, visit, child = deque([v]), [False] * (n + 1), []
visit[v], ma, parent, maxes = True, [0] * (n + 1), [-1] * (n + 1), [[-10 ** 9, 0] for _ in range(n + 1)]
while queue:
s = queue.popleft()
for i1 in self.gdict[s]:
if not visit[i1]:
queue.append(i1)
visit[i1] = True
child.append(i1)
parent[i1] = s
for i in child[::-1]:
ma[parent[i]] = max(ma[i] + 1, ma[parent[i]])
maxes[parent[i]].append(ma[i] + 1)
for i in range(1, n + 1):
maxes[i] = sorted(maxes[i])[-3:]
for i in child:
new = -1
if maxes[parent[i]][-1] == ma[i] + 1:
new = -2
maxes[i].append(maxes[parent[i]][new] + 1)
maxes[i] = sorted(maxes[i])[-3:]
ma[i] = max(ma[i], maxes[parent[i]][new] + 1)
tem = [sum(x) for x in maxes]
ix, ans, su = max(range(n + 1), key=tem.__getitem__), [], 0
visit[ix], mem = False, [() for _ in range(n + 1)]
for i in self.gdict[ix]:
queue.append((i, 1))
visit[i], parent[i] = False, i
while queue:
s, lev = queue.popleft()
flag = 1
for i1 in self.gdict[s]:
if visit[i1]:
queue.append((i1, lev + 1))
visit[i1], flag, parent[i1] = False, 0, parent[s]
if flag:
mem[parent[s]] = (lev, s)
mem.sort()
for i in range(n, n - 3, -1):
if not mem[i]:
break
ans.append(mem[i][1])
su += mem[i][0]
if len(ans) < 3:
ans.append(ix)
print(su, '\n', *ans)
sys.stdin, sys.stdout = IOWrapper(sys.stdin), IOWrapper(sys.stdout)
input = lambda: sys.stdin.readline().rstrip("\r\n")
inp = lambda dtype: [dtype(x) for x in input().split()]
inp_2d = lambda dtype, n: [dtype(input()) for _ in range(n)]
inp_2ds = lambda dtype, n: [inp(dtype) for _ in range(n)]
inp_enu = lambda dtype: [(i, x) for i, x in enumerate(inp(dtype))]
inp_enus = lambda dtype, n: [[i] + inp(dtype) for i in range(n)]
ceil1 = lambda a, b: (a + b - 1) // b
n = int(input())
g = graph(n)
for _ in range(n - 1):
u, v = inp(int)
g.addEdge(u, v)
g.subtree(1)
| py |
1295 | B | B. Infinite Prefixestime limit per test2 secondsmemory limit per test256 megabytesinputstandard inputoutputstandard outputYou are given string ss of length nn consisting of 0-s and 1-s. You build an infinite string tt as a concatenation of an infinite number of strings ss, or t=ssss…t=ssss… For example, if s=s= 10010, then t=t= 100101001010010...Calculate the number of prefixes of tt with balance equal to xx. The balance of some string qq is equal to cnt0,q−cnt1,qcnt0,q−cnt1,q, where cnt0,qcnt0,q is the number of occurrences of 0 in qq, and cnt1,qcnt1,q is the number of occurrences of 1 in qq. The number of such prefixes can be infinite; if it is so, you must say that.A prefix is a string consisting of several first letters of a given string, without any reorders. An empty prefix is also a valid prefix. For example, the string "abcd" has 5 prefixes: empty string, "a", "ab", "abc" and "abcd".InputThe first line contains the single integer TT (1≤T≤1001≤T≤100) — the number of test cases.Next 2T2T lines contain descriptions of test cases — two lines per test case. The first line contains two integers nn and xx (1≤n≤1051≤n≤105, −109≤x≤109−109≤x≤109) — the length of string ss and the desired balance, respectively.The second line contains the binary string ss (|s|=n|s|=n, si∈{0,1}si∈{0,1}).It's guaranteed that the total sum of nn doesn't exceed 105105.OutputPrint TT integers — one per test case. For each test case print the number of prefixes or −1−1 if there is an infinite number of such prefixes.ExampleInputCopy4
6 10
010010
5 3
10101
1 0
0
2 0
01
OutputCopy3
0
1
-1
NoteIn the first test case; there are 3 good prefixes of tt: with length 2828, 3030 and 3232. | [
"math",
"strings"
] | from bisect import *
from collections import *
import sys
import io, os
import math
import random
from heapq import *
gcd = math.gcd
sqrt = math.sqrt
maxint=10**21
def ceil(a, b):
if(b==0):
return maxint
a = -a
k = a // b
k = -k
return k
# arr=list(map(int, input().split()))if(n==1):
# n,m=map(int,input().split())
input = io.BytesIO(os.read(0, os.fstat(0).st_size)).readline
def strinp(testcases):
k = 5
if (testcases == -1 or testcases == 1):
k = 1
f = str(input())
f = f[2:len(f) - k]
return f
def main():
T=int(input())
for _ in range(T):
n,x=map(int,input().split())
s=strinp(90)
z=0
o=0
b=[0]*n
for i in range(n):
if(s[i]=="0"):
z+=1
else:
o+=1
b[i]=z-o
if(b[-1]==0):
trig=False
for i in range(n):
if(b[i]==x):
trig=True
break
if(trig):
print(-1)
else:
print(0)
continue
ans=int((x%b[-1]==0) and ((x//b[-1])>=0))
for i in range(n-1):
g=x-b[i]
if(g%b[-1]==0 and g//b[-1]>=0):
ans+=1
print(ans)
main() | py |
1290 | B | B. Irreducible Anagramstime limit per test2 secondsmemory limit per test256 megabytesinputstandard inputoutputstandard outputLet's call two strings ss and tt anagrams of each other if it is possible to rearrange symbols in the string ss to get a string, equal to tt.Let's consider two strings ss and tt which are anagrams of each other. We say that tt is a reducible anagram of ss if there exists an integer k≥2k≥2 and 2k2k non-empty strings s1,t1,s2,t2,…,sk,tks1,t1,s2,t2,…,sk,tk that satisfy the following conditions: If we write the strings s1,s2,…,sks1,s2,…,sk in order, the resulting string will be equal to ss; If we write the strings t1,t2,…,tkt1,t2,…,tk in order, the resulting string will be equal to tt; For all integers ii between 11 and kk inclusive, sisi and titi are anagrams of each other. If such strings don't exist, then tt is said to be an irreducible anagram of ss. Note that these notions are only defined when ss and tt are anagrams of each other.For example, consider the string s=s= "gamegame". Then the string t=t= "megamage" is a reducible anagram of ss, we may choose for example s1=s1= "game", s2=s2= "gam", s3=s3= "e" and t1=t1= "mega", t2=t2= "mag", t3=t3= "e": On the other hand, we can prove that t=t= "memegaga" is an irreducible anagram of ss.You will be given a string ss and qq queries, represented by two integers 1≤l≤r≤|s|1≤l≤r≤|s| (where |s||s| is equal to the length of the string ss). For each query, you should find if the substring of ss formed by characters from the ll-th to the rr-th has at least one irreducible anagram.InputThe first line contains a string ss, consisting of lowercase English characters (1≤|s|≤2⋅1051≤|s|≤2⋅105).The second line contains a single integer qq (1≤q≤1051≤q≤105) — the number of queries.Each of the following qq lines contain two integers ll and rr (1≤l≤r≤|s|1≤l≤r≤|s|), representing a query for the substring of ss formed by characters from the ll-th to the rr-th.OutputFor each query, print a single line containing "Yes" (without quotes) if the corresponding substring has at least one irreducible anagram, and a single line containing "No" (without quotes) otherwise.ExamplesInputCopyaaaaa
3
1 1
2 4
5 5
OutputCopyYes
No
Yes
InputCopyaabbbbbbc
6
1 2
2 4
2 2
1 9
5 7
3 5
OutputCopyNo
Yes
Yes
Yes
No
No
NoteIn the first sample; in the first and third queries; the substring is "a"; which has itself as an irreducible anagram since two or more non-empty strings cannot be put together to obtain "a". On the other hand; in the second query, the substring is "aaa", which has no irreducible anagrams: its only anagram is itself, and we may choose s1=s1= "a", s2=s2= "aa", t1=t1= "a", t2=t2= "aa" to show that it is a reducible anagram.In the second query of the second sample, the substring is "abb", which has, for example, "bba" as an irreducible anagram. | [
"binary search",
"constructive algorithms",
"data structures",
"strings",
"two pointers"
] | #Code by Sounak, IIESTS
#------------------------------warmup----------------------------
import os
import sys
import math
from io import BytesIO, IOBase
from fractions import Fraction
import collections
from itertools import permutations
from collections import defaultdict
BUFSIZE = 8192
class FastIO(IOBase):
newlines = 0
def __init__(self, file):
self._fd = file.fileno()
self.buffer = BytesIO()
self.writable = "x" in file.mode or "r" not in file.mode
self.write = self.buffer.write if self.writable else None
def read(self):
while True:
b = os.read(self._fd, max(os.fstat(self._fd).st_size, BUFSIZE))
if not b:
break
ptr = self.buffer.tell()
self.buffer.seek(0, 2), self.buffer.write(b), self.buffer.seek(ptr)
self.newlines = 0
return self.buffer.read()
def readline(self):
while self.newlines == 0:
b = os.read(self._fd, max(os.fstat(self._fd).st_size, BUFSIZE))
self.newlines = b.count(b"\n") + (not b)
ptr = self.buffer.tell()
self.buffer.seek(0, 2), self.buffer.write(b), self.buffer.seek(ptr)
self.newlines -= 1
return self.buffer.readline()
def flush(self):
if self.writable:
os.write(self._fd, self.buffer.getvalue())
self.buffer.truncate(0), self.buffer.seek(0)
class IOWrapper(IOBase):
def __init__(self, file):
self.buffer = FastIO(file)
self.flush = self.buffer.flush
self.writable = self.buffer.writable
self.write = lambda s: self.buffer.write(s.encode("ascii"))
self.read = lambda: self.buffer.read().decode("ascii")
self.readline = lambda: self.buffer.readline().decode("ascii")
sys.stdin, sys.stdout = IOWrapper(sys.stdin), IOWrapper(sys.stdout)
input = lambda: sys.stdin.readline().rstrip("\r\n")
#-------------------game starts now-----------------------------------------------------
mod=10**9+7
a=list(input())
n=len(a)
pre=[[0 for j in range (n+1)] for i in range (26)]
for i in range (n):
ch=ord(a[i])-97
for j in range (26):
pre[j][i+1]=pre[j][i]
pre[ch][i+1]+=1
for i in range (int(input())):
l,r=map(int,input().split())
c=0
for j in range (26):
if pre[j][r]!=pre[j][l-1]:
c+=1
#print(pre[j][r]-pre[j][l-1])
if l==r or a[l-1]!=a[r-1] or c>=3:
print("Yes")
else:
print("No") | py |
1141 | G | G. Privatization of Roads in Treelandtime limit per test2 secondsmemory limit per test256 megabytesinputstandard inputoutputstandard outputTreeland consists of nn cities and n−1n−1 roads. Each road is bidirectional and connects two distinct cities. From any city you can get to any other city by roads. Yes, you are right — the country's topology is an undirected tree.There are some private road companies in Treeland. The government decided to sell roads to the companies. Each road will belong to one company and a company can own multiple roads.The government is afraid to look unfair. They think that people in a city can consider them unfair if there is one company which owns two or more roads entering the city. The government wants to make such privatization that the number of such cities doesn't exceed kk and the number of companies taking part in the privatization is minimal.Choose the number of companies rr such that it is possible to assign each road to one company in such a way that the number of cities that have two or more roads of one company is at most kk. In other words, if for a city all the roads belong to the different companies then the city is good. Your task is to find the minimal rr that there is such assignment to companies from 11 to rr that the number of cities which are not good doesn't exceed kk. The picture illustrates the first example (n=6,k=2n=6,k=2). The answer contains r=2r=2 companies. Numbers on the edges denote edge indices. Edge colors mean companies: red corresponds to the first company, blue corresponds to the second company. The gray vertex (number 33) is not good. The number of such vertices (just one) doesn't exceed k=2k=2. It is impossible to have at most k=2k=2 not good cities in case of one company. InputThe first line contains two integers nn and kk (2≤n≤200000,0≤k≤n−12≤n≤200000,0≤k≤n−1) — the number of cities and the maximal number of cities which can have two or more roads belonging to one company.The following n−1n−1 lines contain roads, one road per line. Each line contains a pair of integers xixi, yiyi (1≤xi,yi≤n1≤xi,yi≤n), where xixi, yiyi are cities connected with the ii-th road.OutputIn the first line print the required rr (1≤r≤n−11≤r≤n−1). In the second line print n−1n−1 numbers c1,c2,…,cn−1c1,c2,…,cn−1 (1≤ci≤r1≤ci≤r), where cici is the company to own the ii-th road. If there are multiple answers, print any of them.ExamplesInputCopy6 2
1 4
4 3
3 5
3 6
5 2
OutputCopy2
1 2 1 1 2 InputCopy4 2
3 1
1 4
1 2
OutputCopy1
1 1 1 InputCopy10 2
10 3
1 2
1 3
1 4
2 5
2 6
2 7
3 8
3 9
OutputCopy3
1 1 2 3 2 3 1 3 1 | [
"binary search",
"constructive algorithms",
"dfs and similar",
"graphs",
"greedy",
"trees"
] | from collections import deque; import sys; input = sys.stdin.readline
def bfs(s):
queue = deque([s])
cv = [0] * n
cv[s] = 1
par = [-1] * n
while queue:
u = queue.popleft()
col = 1 + int(par[u] == 1)
for nxt in adj[u]:
if not cv[nxt]:
if u in bad:
if (u, nxt) in d_edges:
d_edges[(u, nxt)] = 1
else:
d_edges[(nxt, u)] = 1
par[nxt] = 1
else:
if (u, nxt) in d_edges:
d_edges[(u, nxt)] = col
else:
d_edges[(nxt, u)] = col
par[nxt] = col
col += 1
if col == par[u]:
col += 1
queue.append(nxt)
cv[nxt] = 1
n, k = [int(item) for item in input().split()]
adj = {}
edges = []
for i in range(n - 1):
u, v = [int(item) - 1 for item in input().split()]
edges.append((u, v))
for j in range(2):
if u in adj: adj[u].append(v)
else: adj[u] = [v]
u, v = v, u
d_edges = {edge: 0 for edge in edges}
p = sorted([(i, len(adj[i])) for i in range(n)], key = lambda p:(p[1], p[0]), reverse=True)
r = p[k][1]
print(r)
bad = set(p[i][0] for i in range(k))
bfs(0)
print(*[d_edges[edge] for edge in edges])
| py |
1141 | F1 | F1. Same Sum Blocks (Easy)time limit per test2 secondsmemory limit per test256 megabytesinputstandard inputoutputstandard outputThis problem is given in two editions; which differ exclusively in the constraints on the number nn.You are given an array of integers a[1],a[2],…,a[n].a[1],a[2],…,a[n]. A block is a sequence of contiguous (consecutive) elements a[l],a[l+1],…,a[r]a[l],a[l+1],…,a[r] (1≤l≤r≤n1≤l≤r≤n). Thus, a block is defined by a pair of indices (l,r)(l,r).Find a set of blocks (l1,r1),(l2,r2),…,(lk,rk)(l1,r1),(l2,r2),…,(lk,rk) such that: They do not intersect (i.e. they are disjoint). Formally, for each pair of blocks (li,ri)(li,ri) and (lj,rj(lj,rj) where i≠ji≠j either ri<ljri<lj or rj<lirj<li. For each block the sum of its elements is the same. Formally, a[l1]+a[l1+1]+⋯+a[r1]=a[l2]+a[l2+1]+⋯+a[r2]=a[l1]+a[l1+1]+⋯+a[r1]=a[l2]+a[l2+1]+⋯+a[r2]= ⋯=⋯= a[lk]+a[lk+1]+⋯+a[rk].a[lk]+a[lk+1]+⋯+a[rk]. The number of the blocks in the set is maximum. Formally, there does not exist a set of blocks (l′1,r′1),(l′2,r′2),…,(l′k′,r′k′)(l1′,r1′),(l2′,r2′),…,(lk′′,rk′′) satisfying the above two requirements with k′>kk′>k. The picture corresponds to the first example. Blue boxes illustrate blocks. Write a program to find such a set of blocks.InputThe first line contains integer nn (1≤n≤501≤n≤50) — the length of the given array. The second line contains the sequence of elements a[1],a[2],…,a[n]a[1],a[2],…,a[n] (−105≤ai≤105−105≤ai≤105).OutputIn the first line print the integer kk (1≤k≤n1≤k≤n). The following kk lines should contain blocks, one per line. In each line print a pair of indices li,rili,ri (1≤li≤ri≤n1≤li≤ri≤n) — the bounds of the ii-th block. You can print blocks in any order. If there are multiple answers, print any of them.ExamplesInputCopy7
4 1 2 2 1 5 3
OutputCopy3
7 7
2 3
4 5
InputCopy11
-5 -4 -3 -2 -1 0 1 2 3 4 5
OutputCopy2
3 4
1 1
InputCopy4
1 1 1 1
OutputCopy4
4 4
1 1
2 2
3 3
| [
"greedy"
] | import sys, os, io
input = lambda: sys.stdin.readline().rstrip('\r\n')
from collections import defaultdict
n = int(input())
a = list(map(int,input().split()))
has = defaultdict(list)
for j in range(n):
lin = 0
for i in range(j,-1,-1):
lin += a[i]
if lin in has and has[lin][-1]%n >= i:continue
has[lin].append(i*n+j)
mx = max(has.values(),key=lambda x:len(x))
print(len(mx))
for i in mx:
print(i//n+1,i%n+1) | py |
1315 | B | B. Homecomingtime limit per test1 secondmemory limit per test256 megabytesinputstandard inputoutputstandard outputAfter a long party Petya decided to return home; but he turned out to be at the opposite end of the town from his home. There are nn crossroads in the line in the town, and there is either the bus or the tram station at each crossroad.The crossroads are represented as a string ss of length nn, where si=Asi=A, if there is a bus station at ii-th crossroad, and si=Bsi=B, if there is a tram station at ii-th crossroad. Currently Petya is at the first crossroad (which corresponds to s1s1) and his goal is to get to the last crossroad (which corresponds to snsn).If for two crossroads ii and jj for all crossroads i,i+1,…,j−1i,i+1,…,j−1 there is a bus station, one can pay aa roubles for the bus ticket, and go from ii-th crossroad to the jj-th crossroad by the bus (it is not necessary to have a bus station at the jj-th crossroad). Formally, paying aa roubles Petya can go from ii to jj if st=Ast=A for all i≤t<ji≤t<j. If for two crossroads ii and jj for all crossroads i,i+1,…,j−1i,i+1,…,j−1 there is a tram station, one can pay bb roubles for the tram ticket, and go from ii-th crossroad to the jj-th crossroad by the tram (it is not necessary to have a tram station at the jj-th crossroad). Formally, paying bb roubles Petya can go from ii to jj if st=Bst=B for all i≤t<ji≤t<j.For example, if ss="AABBBAB", a=4a=4 and b=3b=3 then Petya needs: buy one bus ticket to get from 11 to 33, buy one tram ticket to get from 33 to 66, buy one bus ticket to get from 66 to 77. Thus, in total he needs to spend 4+3+4=114+3+4=11 roubles. Please note that the type of the stop at the last crossroad (i.e. the character snsn) does not affect the final expense.Now Petya is at the first crossroad, and he wants to get to the nn-th crossroad. After the party he has left with pp roubles. He's decided to go to some station on foot, and then go to home using only public transport.Help him to choose the closest crossroad ii to go on foot the first, so he has enough money to get from the ii-th crossroad to the nn-th, using only tram and bus tickets.InputEach test contains one or more test cases. The first line contains the number of test cases tt (1≤t≤1041≤t≤104).The first line of each test case consists of three integers a,b,pa,b,p (1≤a,b,p≤1051≤a,b,p≤105) — the cost of bus ticket, the cost of tram ticket and the amount of money Petya has.The second line of each test case consists of one string ss, where si=Asi=A, if there is a bus station at ii-th crossroad, and si=Bsi=B, if there is a tram station at ii-th crossroad (2≤|s|≤1052≤|s|≤105).It is guaranteed, that the sum of the length of strings ss by all test cases in one test doesn't exceed 105105.OutputFor each test case print one number — the minimal index ii of a crossroad Petya should go on foot. The rest of the path (i.e. from ii to nn he should use public transport).ExampleInputCopy5
2 2 1
BB
1 1 1
AB
3 2 8
AABBBBAABB
5 3 4
BBBBB
2 1 1
ABABAB
OutputCopy2
1
3
1
6
| [
"binary search",
"dp",
"greedy",
"strings"
] | t = int(input())
for tt in range(t):
a, b, p = map(int, input().split())
s = input()
cs = {'A':a, 'B':b}
c = 0
i = len(s)-1
while i > 0 and c+cs[s[i-1]] <= p:
# print(tt, i)
c += cs[s[i-1]]
i -= 1
while i > 0 and s[i-1] == s[i]:
i -= 1
print(i+1)
# B. Homecoming | py |
1141 | C | C. Polycarp Restores Permutationtime limit per test2 secondsmemory limit per test256 megabytesinputstandard inputoutputstandard outputAn array of integers p1,p2,…,pnp1,p2,…,pn is called a permutation if it contains each number from 11 to nn exactly once. For example, the following arrays are permutations: [3,1,2][3,1,2], [1][1], [1,2,3,4,5][1,2,3,4,5] and [4,3,1,2][4,3,1,2]. The following arrays are not permutations: [2][2], [1,1][1,1], [2,3,4][2,3,4].Polycarp invented a really cool permutation p1,p2,…,pnp1,p2,…,pn of length nn. It is very disappointing, but he forgot this permutation. He only remembers the array q1,q2,…,qn−1q1,q2,…,qn−1 of length n−1n−1, where qi=pi+1−piqi=pi+1−pi.Given nn and q=q1,q2,…,qn−1q=q1,q2,…,qn−1, help Polycarp restore the invented permutation.InputThe first line contains the integer nn (2≤n≤2⋅1052≤n≤2⋅105) — the length of the permutation to restore. The second line contains n−1n−1 integers q1,q2,…,qn−1q1,q2,…,qn−1 (−n<qi<n−n<qi<n).OutputPrint the integer -1 if there is no such permutation of length nn which corresponds to the given array qq. Otherwise, if it exists, print p1,p2,…,pnp1,p2,…,pn. Print any such permutation if there are many of them.ExamplesInputCopy3
-2 1
OutputCopy3 1 2 InputCopy5
1 1 1 1
OutputCopy1 2 3 4 5 InputCopy4
-1 2 2
OutputCopy-1
| [
"math"
] | n = int(input())
a = list(map(int,input().split()))
lst=[ ]
s = 0
flag = 0
for i in a:
s+=i
flag = min(flag,s)
lst.append(1-flag)
for i in range(1,n):
lst.append(lst[-1]+a[i-1])
d = dict()
for i in lst:
d[i]=0
xy=0
for i in lst:
d[i]+=1
if (d[i]>1):
xy=1
if (min(lst)<1 or max(lst)>n or xy==1):
print(-1)
else:
print(*lst)
| py |
1296 | B | B. Food Buyingtime limit per test1 secondmemory limit per test256 megabytesinputstandard inputoutputstandard outputMishka wants to buy some food in the nearby shop. Initially; he has ss burles on his card. Mishka can perform the following operation any number of times (possibly, zero): choose some positive integer number 1≤x≤s1≤x≤s, buy food that costs exactly xx burles and obtain ⌊x10⌋⌊x10⌋ burles as a cashback (in other words, Mishka spends xx burles and obtains ⌊x10⌋⌊x10⌋ back). The operation ⌊ab⌋⌊ab⌋ means aa divided by bb rounded down.It is guaranteed that you can always buy some food that costs xx for any possible value of xx.Your task is to say the maximum number of burles Mishka can spend if he buys food optimally.For example, if Mishka has s=19s=19 burles then the maximum number of burles he can spend is 2121. Firstly, he can spend x=10x=10 burles, obtain 11 burle as a cashback. Now he has s=10s=10 burles, so can spend x=10x=10 burles, obtain 11 burle as a cashback and spend it too.You have to answer tt independent test cases.InputThe first line of the input contains one integer tt (1≤t≤1041≤t≤104) — the number of test cases.The next tt lines describe test cases. Each test case is given on a separate line and consists of one integer ss (1≤s≤1091≤s≤109) — the number of burles Mishka initially has.OutputFor each test case print the answer on it — the maximum number of burles Mishka can spend if he buys food optimally.ExampleInputCopy6
1
10
19
9876
12345
1000000000
OutputCopy1
11
21
10973
13716
1111111111
| [
"math"
] | t = int(input())
for i in range(t):
s = int(input())
sum1 = 0
while s > 9:
x = s - (s % 10)
s = s - x + x // 10
sum1 += x
sum1 += s
print(sum1) | py |
1299 | A | A. Anu Has a Functiontime limit per test1 secondmemory limit per test256 megabytesinputstandard inputoutputstandard outputAnu has created her own function ff: f(x,y)=(x|y)−yf(x,y)=(x|y)−y where || denotes the bitwise OR operation. For example, f(11,6)=(11|6)−6=15−6=9f(11,6)=(11|6)−6=15−6=9. It can be proved that for any nonnegative numbers xx and yy value of f(x,y)f(x,y) is also nonnegative. She would like to research more about this function and has created multiple problems for herself. But she isn't able to solve all of them and needs your help. Here is one of these problems.A value of an array [a1,a2,…,an][a1,a2,…,an] is defined as f(f(…f(f(a1,a2),a3),…an−1),an)f(f(…f(f(a1,a2),a3),…an−1),an) (see notes). You are given an array with not necessarily distinct elements. How should you reorder its elements so that the value of the array is maximal possible?InputThe first line contains a single integer nn (1≤n≤1051≤n≤105).The second line contains nn integers a1,a2,…,ana1,a2,…,an (0≤ai≤1090≤ai≤109). Elements of the array are not guaranteed to be different.OutputOutput nn integers, the reordering of the array with maximum value. If there are multiple answers, print any.ExamplesInputCopy4
4 0 11 6
OutputCopy11 6 4 0InputCopy1
13
OutputCopy13 NoteIn the first testcase; value of the array [11,6,4,0][11,6,4,0] is f(f(f(11,6),4),0)=f(f(9,4),0)=f(9,0)=9f(f(f(11,6),4),0)=f(f(9,4),0)=f(9,0)=9.[11,4,0,6][11,4,0,6] is also a valid answer. | [
"brute force",
"greedy",
"math"
] |
def solve():
num_values = int(input())
values = [int(each) for each in input().split()]
if num_values == 1:
print(values[0])
return
right_and = [values[0]]
left_and = [values[-1]]
for i in range(1, num_values):
right_and.append(values[i]|right_and[-1])
for i in range(num_values-2, -1, -1):
left_and.append(values[i]|left_and[-1])
left_and = list(reversed(left_and))
best = values[0]&(~left_and[1])
best_value = values[0]
right = values[-1]&(~right_and[-2])
if right > best:
best = right
best_value = values[-1]
for i in range(1, num_values - 1):
curr_value = values[i]
curr_best = curr_value&(~(right_and[i-1]|left_and[i+1]))
if curr_best > best:
best = curr_best
best_value = curr_value
new_values = [best_value]
flag = False
for each in values:
if flag or each != best_value:
new_values.append(each)
else:
flag = True
print(" ".join([str(each) for each in new_values]))
solve() | py |
1315 | A | A. Dead Pixeltime limit per test1 secondmemory limit per test256 megabytesinputstandard inputoutputstandard outputScreen resolution of Polycarp's monitor is a×ba×b pixels. Unfortunately, there is one dead pixel at his screen. It has coordinates (x,y)(x,y) (0≤x<a,0≤y<b0≤x<a,0≤y<b). You can consider columns of pixels to be numbered from 00 to a−1a−1, and rows — from 00 to b−1b−1.Polycarp wants to open a rectangular window of maximal size, which doesn't contain the dead pixel. The boundaries of the window should be parallel to the sides of the screen.Print the maximal area (in pixels) of a window that doesn't contain the dead pixel inside itself.InputIn the first line you are given an integer tt (1≤t≤1041≤t≤104) — the number of test cases in the test. In the next lines you are given descriptions of tt test cases.Each test case contains a single line which consists of 44 integers a,b,xa,b,x and yy (1≤a,b≤1041≤a,b≤104; 0≤x<a0≤x<a; 0≤y<b0≤y<b) — the resolution of the screen and the coordinates of a dead pixel. It is guaranteed that a+b>2a+b>2 (e.g. a=b=1a=b=1 is impossible).OutputPrint tt integers — the answers for each test case. Each answer should contain an integer equal to the maximal possible area (in pixels) of a rectangular window, that doesn't contain the dead pixel.ExampleInputCopy6
8 8 0 0
1 10 0 3
17 31 10 4
2 1 0 0
5 10 3 9
10 10 4 8
OutputCopy56
6
442
1
45
80
NoteIn the first test case; the screen resolution is 8×88×8, and the upper left pixel is a dead pixel. Here you can see one of two possible layouts of the maximal window. | [
"implementation"
] | L=[]
def fonction(a,b,x,y):
pr1=b*x
pr2=b*(a-x-1)
pr3=a*y
pr4=a*(b-y-1)
return max(pr1,pr2,pr3,pr4)
t=int(input())
for i in range(t):
a,b,x,y=[int(x) for x in input().split()]
L.append(fonction(a,b,x,y))
for k in L:
print(k) | py |
1296 | A | A. Array with Odd Sumtime limit per test1 secondmemory limit per test256 megabytesinputstandard inputoutputstandard outputYou are given an array aa consisting of nn integers.In one move, you can choose two indices 1≤i,j≤n1≤i,j≤n such that i≠ji≠j and set ai:=ajai:=aj. You can perform such moves any number of times (possibly, zero). You can choose different indices in different operations. The operation := is the operation of assignment (i.e. you choose ii and jj and replace aiai with ajaj).Your task is to say if it is possible to obtain an array with an odd (not divisible by 22) sum of elements.You have to answer tt independent test cases.InputThe first line of the input contains one integer tt (1≤t≤20001≤t≤2000) — the number of test cases.The next 2t2t lines describe test cases. The first line of the test case contains one integer nn (1≤n≤20001≤n≤2000) — the number of elements in aa. The second line of the test case contains nn integers a1,a2,…,ana1,a2,…,an (1≤ai≤20001≤ai≤2000), where aiai is the ii-th element of aa.It is guaranteed that the sum of nn over all test cases does not exceed 20002000 (∑n≤2000∑n≤2000).OutputFor each test case, print the answer on it — "YES" (without quotes) if it is possible to obtain the array with an odd sum of elements, and "NO" otherwise.ExampleInputCopy5
2
2 3
4
2 2 8 8
3
3 3 3
4
5 5 5 5
4
1 1 1 1
OutputCopyYES
NO
YES
NO
NO
| [
"math"
] | t=int(input())
for i in range(t):
n=int(input())
m=input().split()
a=[int(i) for i in m]
odd=0
even=0
for j in a:
if j%2==0:
even+=j
else:
odd+=j
if sum(a)%2==1:
print("YES")
else:
if odd>1 and even>1:
print("YES")
else:
print("NO") | py |
1303 | A | A. Erasing Zeroestime limit per test1 secondmemory limit per test256 megabytesinputstandard inputoutputstandard outputYou are given a string ss. Each character is either 0 or 1.You want all 1's in the string to form a contiguous subsegment. For example, if the string is 0, 1, 00111 or 01111100, then all 1's form a contiguous subsegment, and if the string is 0101, 100001 or 11111111111101, then this condition is not met.You may erase some (possibly none) 0's from the string. What is the minimum number of 0's that you have to erase?InputThe first line contains one integer tt (1≤t≤1001≤t≤100) — the number of test cases.Then tt lines follow, each representing a test case. Each line contains one string ss (1≤|s|≤1001≤|s|≤100); each character of ss is either 0 or 1.OutputPrint tt integers, where the ii-th integer is the answer to the ii-th testcase (the minimum number of 0's that you have to erase from ss).ExampleInputCopy3
010011
0
1111000
OutputCopy2
0
0
NoteIn the first test case you have to delete the third and forth symbols from string 010011 (it turns into 0111). | [
"implementation",
"strings"
] | for i in range(int(input())):
s = input()
while len(s) > 0 and s[0] == "0":
s = s[1:]
while len(s) > 0 and s[-1] == "0":
s = s[:-1]
res = 0
for x in s:
if x == "0":
res += 1
print(res) | py |
1301 | C | C. Ayoub's functiontime limit per test1 secondmemory limit per test256 megabytesinputstandard inputoutputstandard outputAyoub thinks that he is a very smart person; so he created a function f(s)f(s), where ss is a binary string (a string which contains only symbols "0" and "1"). The function f(s)f(s) is equal to the number of substrings in the string ss that contains at least one symbol, that is equal to "1".More formally, f(s)f(s) is equal to the number of pairs of integers (l,r)(l,r), such that 1≤l≤r≤|s|1≤l≤r≤|s| (where |s||s| is equal to the length of string ss), such that at least one of the symbols sl,sl+1,…,srsl,sl+1,…,sr is equal to "1". For example, if s=s="01010" then f(s)=12f(s)=12, because there are 1212 such pairs (l,r)(l,r): (1,2),(1,3),(1,4),(1,5),(2,2),(2,3),(2,4),(2,5),(3,4),(3,5),(4,4),(4,5)(1,2),(1,3),(1,4),(1,5),(2,2),(2,3),(2,4),(2,5),(3,4),(3,5),(4,4),(4,5).Ayoub also thinks that he is smarter than Mahmoud so he gave him two integers nn and mm and asked him this problem. For all binary strings ss of length nn which contains exactly mm symbols equal to "1", find the maximum value of f(s)f(s).Mahmoud couldn't solve the problem so he asked you for help. Can you help him? InputThe input consists of multiple test cases. The first line contains a single integer tt (1≤t≤1051≤t≤105) — the number of test cases. The description of the test cases follows.The only line for each test case contains two integers nn, mm (1≤n≤1091≤n≤109, 0≤m≤n0≤m≤n) — the length of the string and the number of symbols equal to "1" in it.OutputFor every test case print one integer number — the maximum value of f(s)f(s) over all strings ss of length nn, which has exactly mm symbols, equal to "1".ExampleInputCopy5
3 1
3 2
3 3
4 0
5 2
OutputCopy4
5
6
0
12
NoteIn the first test case; there exists only 33 strings of length 33, which has exactly 11 symbol, equal to "1". These strings are: s1=s1="100", s2=s2="010", s3=s3="001". The values of ff for them are: f(s1)=3,f(s2)=4,f(s3)=3f(s1)=3,f(s2)=4,f(s3)=3, so the maximum value is 44 and the answer is 44.In the second test case, the string ss with the maximum value is "101".In the third test case, the string ss with the maximum value is "111".In the fourth test case, the only string ss of length 44, which has exactly 00 symbols, equal to "1" is "0000" and the value of ff for that string is 00, so the answer is 00.In the fifth test case, the string ss with the maximum value is "01010" and it is described as an example in the problem statement. | [
"binary search",
"combinatorics",
"greedy",
"math",
"strings"
] | # 2022-09-04 15:27:04.802963
# https://codeforces.com/problemset/problem/1301/C
import sys
_DEBUG = False
# _DEBUG = True
if not _DEBUG:
input = sys.stdin.readline
print = sys.stdout.write
"""
6 2
7 // 3 == 2
001001 3 + 3
001010 3 + 1 + 1
"""
def proc(n, m):
total_num = n * (n + 1) // 2
num_of_zero = n - m
group_count = m + 1
num_per_group = num_of_zero // group_count
remains = num_of_zero % group_count
return total_num - num_per_group * (num_per_group + 1) // 2 * group_count - remains * (num_per_group + 1)
if _DEBUG:
assert proc(3, 1) == 4
assert proc(2, 1) == 2
assert proc(5, 1) == 9
assert proc(3, 3) == 6
assert proc(4, 0) == 0
assert proc(5, 2) == 12
# fin = open('input.txt')
# fou = open('output.txt')
#
# fin.readline()
# for line in fin.readlines():
# n, m = map(int, line.split())
#
# expected = int(fou.readline())
#
# if proc(n, m) != expected:
# print('proc({}, {}) != {} !!'.format(n, m, expected))
for _ in range(int(input())):
n, m = map(int, input().split())
ans = proc(n, m)
if not _DEBUG:
print(str(ans) + '\n')
else:
print(ans)
| py |
1299 | A | A. Anu Has a Functiontime limit per test1 secondmemory limit per test256 megabytesinputstandard inputoutputstandard outputAnu has created her own function ff: f(x,y)=(x|y)−yf(x,y)=(x|y)−y where || denotes the bitwise OR operation. For example, f(11,6)=(11|6)−6=15−6=9f(11,6)=(11|6)−6=15−6=9. It can be proved that for any nonnegative numbers xx and yy value of f(x,y)f(x,y) is also nonnegative. She would like to research more about this function and has created multiple problems for herself. But she isn't able to solve all of them and needs your help. Here is one of these problems.A value of an array [a1,a2,…,an][a1,a2,…,an] is defined as f(f(…f(f(a1,a2),a3),…an−1),an)f(f(…f(f(a1,a2),a3),…an−1),an) (see notes). You are given an array with not necessarily distinct elements. How should you reorder its elements so that the value of the array is maximal possible?InputThe first line contains a single integer nn (1≤n≤1051≤n≤105).The second line contains nn integers a1,a2,…,ana1,a2,…,an (0≤ai≤1090≤ai≤109). Elements of the array are not guaranteed to be different.OutputOutput nn integers, the reordering of the array with maximum value. If there are multiple answers, print any.ExamplesInputCopy4
4 0 11 6
OutputCopy11 6 4 0InputCopy1
13
OutputCopy13 NoteIn the first testcase; value of the array [11,6,4,0][11,6,4,0] is f(f(f(11,6),4),0)=f(f(9,4),0)=f(9,0)=9f(f(f(11,6),4),0)=f(f(9,4),0)=f(9,0)=9.[11,4,0,6][11,4,0,6] is also a valid answer. | [
"brute force",
"greedy",
"math"
] | n = int(input())
a = list(map(int,input().split()))
bits = [0]*30
for i in a:
x = 1
for j in range(30):
if i&x:
bits[j] += 1
x *= 2
z = -1
for i in range(29,-1,-1):
if bits[i] == 1:
z = i
break
if z >= 0:
x = 2**z
for i in a:
if x&i:
break
l = [i]
for j in a:
if j != i:
l.append(j)
print(*l)
else:
print(*a) | py |
1312 | G | G. Autocompletiontime limit per test7 secondsmemory limit per test512 megabytesinputstandard inputoutputstandard outputYou are given a set of strings SS. Each string consists of lowercase Latin letters.For each string in this set, you want to calculate the minimum number of seconds required to type this string. To type a string, you have to start with an empty string and transform it into the string you want to type using the following actions: if the current string is tt, choose some lowercase Latin letter cc and append it to the back of tt, so the current string becomes t+ct+c. This action takes 11 second; use autocompletion. When you try to autocomplete the current string tt, a list of all strings s∈Ss∈S such that tt is a prefix of ss is shown to you. This list includes tt itself, if tt is a string from SS, and the strings are ordered lexicographically. You can transform tt into the ii-th string from this list in ii seconds. Note that you may choose any string from this list you want, it is not necessarily the string you are trying to type. What is the minimum number of seconds that you have to spend to type each string from SS?Note that the strings from SS are given in an unusual way.InputThe first line contains one integer nn (1≤n≤1061≤n≤106).Then nn lines follow, the ii-th line contains one integer pipi (0≤pi<i0≤pi<i) and one lowercase Latin character cici. These lines form some set of strings such that SS is its subset as follows: there are n+1n+1 strings, numbered from 00 to nn; the 00-th string is an empty string, and the ii-th string (i≥1i≥1) is the result of appending the character cici to the string pipi. It is guaranteed that all these strings are distinct.The next line contains one integer kk (1≤k≤n1≤k≤n) — the number of strings in SS.The last line contains kk integers a1a1, a2a2, ..., akak (1≤ai≤n1≤ai≤n, all aiai are pairwise distinct) denoting the indices of the strings generated by above-mentioned process that form the set SS — formally, if we denote the ii-th generated string as sisi, then S=sa1,sa2,…,sakS=sa1,sa2,…,sak.OutputPrint kk integers, the ii-th of them should be equal to the minimum number of seconds required to type the string saisai.ExamplesInputCopy10
0 i
1 q
2 g
0 k
1 e
5 r
4 m
5 h
3 p
3 e
5
8 9 1 10 6
OutputCopy2 4 1 3 3
InputCopy8
0 a
1 b
2 a
2 b
4 a
4 b
5 c
6 d
5
2 3 4 7 8
OutputCopy1 2 2 4 4
NoteIn the first example; SS consists of the following strings: ieh, iqgp, i, iqge, ier. | [
"data structures",
"dfs and similar",
"dp"
] | import sys
input = sys.stdin.readline
n=int(input())
T=[input().split() for i in range(n)]
k=int(input())
S=list(map(int,input().split()))
SETS=set(S)
E=[[] for i in range(n+1)]
P=[-1]*(n+1)
for i in range(n):
p,s=T[i]
p=int(p)
E[p].append((s,i+1))
P[i+1]=p
for i in range(n+1):
E[i].sort(reverse=True)
ELI=[0]*(n+1)
DEPTH=[0]*(n+1)
ELIMIN=[0]*(n+1)
ANS=[0]*(n+1)
Q=[0]
USED=[0]*(n+1)
count=0
while Q:
x=Q.pop()
USED[x]=1
if x in SETS:
count+=1
#print(x,count)
if x in SETS:
ANS[x]=min(DEPTH[x],count+ELIMIN[P[x]],ANS[P[x]]+1)
ELI[x]=ANS[x]-count+1
else:
ANS[x]=min(DEPTH[x],ANS[P[x]]+1)
ELI[x]=ANS[x]-count
ELIMIN[x]=min(ELI[x],ELIMIN[P[x]])
for s,to in E[x]:
if USED[to]==1:
continue
Q.append(to)
DEPTH[to]=DEPTH[x]+1
print(*[ANS[s] for s in S])
| py |
1325 | E | E. Ehab's REAL Number Theory Problemtime limit per test3 secondsmemory limit per test256 megabytesinputstandard inputoutputstandard outputYou are given an array aa of length nn that has a special condition: every element in this array has at most 7 divisors. Find the length of the shortest non-empty subsequence of this array product of whose elements is a perfect square.A sequence aa is a subsequence of an array bb if aa can be obtained from bb by deletion of several (possibly, zero or all) elements.InputThe first line contains an integer nn (1≤n≤1051≤n≤105) — the length of aa.The second line contains nn integers a1a1, a2a2, ……, anan (1≤ai≤1061≤ai≤106) — the elements of the array aa.OutputOutput the length of the shortest non-empty subsequence of aa product of whose elements is a perfect square. If there are several shortest subsequences, you can find any of them. If there's no such subsequence, print "-1".ExamplesInputCopy3
1 4 6
OutputCopy1InputCopy4
2 3 6 6
OutputCopy2InputCopy3
6 15 10
OutputCopy3InputCopy4
2 3 5 7
OutputCopy-1NoteIn the first sample; you can choose a subsequence [1][1].In the second sample, you can choose a subsequence [6,6][6,6].In the third sample, you can choose a subsequence [6,15,10][6,15,10].In the fourth sample, there is no such subsequence. | [
"brute force",
"dfs and similar",
"graphs",
"number theory",
"shortest paths"
] | import sys
MAX = 1_000_005
min_p_factor = [0] * MAX
pr = []
pid = {1: 0}
for i in range(2, MAX):
if not min_p_factor[i]:
min_p_factor[i] = i
pr.append(i)
pid[i] = len(pr)
for p in pr:
if p > min_p_factor[i] or i * p >= MAX:
break
min_p_factor[i * p] = p
n = int(input())
a = list(map(int, input().split()))
g = [[] for _ in range(len(pid))]
for x in a:
f = []
while x > 1:
p, c = min_p_factor[x], 0
while min_p_factor[x] == p:
x //= p
c ^= 1
if c:
f.append(p)
if not f:
print(1)
exit()
f.append(1)
u, v = pid[f[0]], pid[f[1]]
g[u].append(v)
g[v].append(u)
def shortest_cycle(s):
depth = [-1] * len(pid)
depth[s] = 0
q = [(s, -1)]
while q:
q2 = []
for u, p in q:
for v in g[u]:
if depth[v] != -1:
if v != p:
return depth[u] + depth[v] + 1
else:
depth[v] = depth[u] + 1
q2.append((v, u))
q = q2
shortest = n + 1
for i in range(169): # pid[1009] == 169
shortest = min(shortest, shortest_cycle(i) or shortest)
print(shortest if shortest <= n else -1)
| py |
1315 | C | C. Restoring Permutationtime limit per test1 secondmemory limit per test256 megabytesinputstandard inputoutputstandard outputYou are given a sequence b1,b2,…,bnb1,b2,…,bn. Find the lexicographically minimal permutation a1,a2,…,a2na1,a2,…,a2n such that bi=min(a2i−1,a2i)bi=min(a2i−1,a2i), or determine that it is impossible.InputEach test contains one or more test cases. The first line contains the number of test cases tt (1≤t≤1001≤t≤100).The first line of each test case consists of one integer nn — the number of elements in the sequence bb (1≤n≤1001≤n≤100).The second line of each test case consists of nn different integers b1,…,bnb1,…,bn — elements of the sequence bb (1≤bi≤2n1≤bi≤2n).It is guaranteed that the sum of nn by all test cases doesn't exceed 100100.OutputFor each test case, if there is no appropriate permutation, print one number −1−1.Otherwise, print 2n2n integers a1,…,a2na1,…,a2n — required lexicographically minimal permutation of numbers from 11 to 2n2n.ExampleInputCopy5
1
1
2
4 1
3
4 1 3
4
2 3 4 5
5
1 5 7 2 8
OutputCopy1 2
-1
4 5 1 2 3 6
-1
1 3 5 6 7 9 2 4 8 10
| [
"greedy"
] | from sys import stdin
input = stdin.readline
#google = lambda : print("Case #%d: "%(T + 1) , end = '')
inp = lambda : list(map(int,input().split()))
def answer():
cantake = set([i for i in range(1 , 2 * n + 1)])
ans = [-1 for i in range(2 * n)]
for i in range(n):
ans[2 * i] = a[i]
cantake.remove(a[i])
for i in range(n):
ok = False
for j in range(a[i] , 2 * n + 1):
if(j in cantake):
ans[2 * i + 1] = j
cantake.remove(j)
ok = True
break
if(not ok):return [-1]
return ans
for T in range(int(input())):
n = int(input())
a = inp()
print(*answer())
| py |
1141 | F2 | F2. Same Sum Blocks (Hard)time limit per test3 secondsmemory limit per test256 megabytesinputstandard inputoutputstandard outputThis problem is given in two editions; which differ exclusively in the constraints on the number nn.You are given an array of integers a[1],a[2],…,a[n].a[1],a[2],…,a[n]. A block is a sequence of contiguous (consecutive) elements a[l],a[l+1],…,a[r]a[l],a[l+1],…,a[r] (1≤l≤r≤n1≤l≤r≤n). Thus, a block is defined by a pair of indices (l,r)(l,r).Find a set of blocks (l1,r1),(l2,r2),…,(lk,rk)(l1,r1),(l2,r2),…,(lk,rk) such that: They do not intersect (i.e. they are disjoint). Formally, for each pair of blocks (li,ri)(li,ri) and (lj,rj(lj,rj) where i≠ji≠j either ri<ljri<lj or rj<lirj<li. For each block the sum of its elements is the same. Formally, a[l1]+a[l1+1]+⋯+a[r1]=a[l2]+a[l2+1]+⋯+a[r2]=a[l1]+a[l1+1]+⋯+a[r1]=a[l2]+a[l2+1]+⋯+a[r2]= ⋯=⋯= a[lk]+a[lk+1]+⋯+a[rk].a[lk]+a[lk+1]+⋯+a[rk]. The number of the blocks in the set is maximum. Formally, there does not exist a set of blocks (l′1,r′1),(l′2,r′2),…,(l′k′,r′k′)(l1′,r1′),(l2′,r2′),…,(lk′′,rk′′) satisfying the above two requirements with k′>kk′>k. The picture corresponds to the first example. Blue boxes illustrate blocks. Write a program to find such a set of blocks.InputThe first line contains integer nn (1≤n≤15001≤n≤1500) — the length of the given array. The second line contains the sequence of elements a[1],a[2],…,a[n]a[1],a[2],…,a[n] (−105≤ai≤105−105≤ai≤105).OutputIn the first line print the integer kk (1≤k≤n1≤k≤n). The following kk lines should contain blocks, one per line. In each line print a pair of indices li,rili,ri (1≤li≤ri≤n1≤li≤ri≤n) — the bounds of the ii-th block. You can print blocks in any order. If there are multiple answers, print any of them.ExamplesInputCopy7
4 1 2 2 1 5 3
OutputCopy3
7 7
2 3
4 5
InputCopy11
-5 -4 -3 -2 -1 0 1 2 3 4 5
OutputCopy2
3 4
1 1
InputCopy4
1 1 1 1
OutputCopy4
4 4
1 1
2 2
3 3
| [
"data structures",
"greedy"
] | # Problem: F2. Same Sum Blocks (Hard)
# Contest: Codeforces - Codeforces Round #547 (Div. 3)
# URL: https://codeforces.com/problemset/problem/1141/F2
# Memory Limit: 256 MB
# Time Limit: 3000 ms
import sys
import bisect
import random
import io, os
from bisect import *
from collections import *
from contextlib import redirect_stdout
from itertools import *
from array import *
from functools import lru_cache
from types import GeneratorType
from heapq import *
from math import sqrt, gcd, inf
if sys.version >= '3.8': # ACW没有comb
from math import comb
RI = lambda: map(int, sys.stdin.buffer.readline().split())
RS = lambda: map(bytes.decode, sys.stdin.buffer.readline().strip().split())
RILST = lambda: list(RI())
DEBUG = lambda *x: sys.stderr.write(f'{str(x)}\n')
MOD = 10 ** 9 + 7
PROBLEM = """https://codeforces.com/problemset/problem/1141/F2
输入 n(≤1500) 和长为 n 的数组 a(-1e5≤a[i]≤1e5)，下标从 1 开始。
你需要从 a 中选出尽量多的非空连续子数组，这些子数组不能重叠，且元素和相等。
输出子数组的个数 k，然后输出 k 行，每行两个数表示子数组的左右端点。
可以按任意顺序输出，多种方案可以输出任意一种。
"""
"""https://codeforces.com/contest/1141/submission/192610240
暴力统计每个子数组的和，用哈希表把和相同的子数组左右端点记录下来。
对于每一组，问题变成最多不重叠线段个数。
这是个经典贪心，按照右端点从小到大排序+遍历，一旦遇到左端点大于上一个记录的右端点，答案加一，更新右端点。"""
# 1091 ms 暴力+右端点排序+贪心 n^2lgn
def solve1():
n, = RI()
a = RILST()
d = defaultdict(list)
for i in range(n):
s = 0
for j in range(i, n):
s += a[j]
d[s].append((i + 1, j + 1))
ans = []
for lst in d.values():
if len(lst) <= len(ans):
continue
lst.sort(key=lambda x: x[1])
t = []
r0 = 0
for l, r in lst:
if l > r0:
t.append((l, r))
r0 = r
if len(t) > len(ans):
ans = t
print(len(ans))
print('\n'.join(map(lambda x: f'{x[0]} {x[1]}', ans)))
# 1122 ms 暴力+左端点排序(省去)+贪心 n^2
def solve2():
n, = RI()
a = RILST()
d = defaultdict(list)
for i in range(n):
s = 0
for j in range(i, n):
s += a[j]
d[s].append((i + 1, j + 1))
ans = []
for lst in d.values():
if len(lst) <= len(ans):
continue
t = []
l1 = 10 ** 6
for l, r in lst[::-1]:
if r < l1:
t.append((l, r))
l1 = l
if len(t) > len(ans):
ans = t
print(len(ans))
print('\n'.join(map(lambda x: f'{x[0]} {x[1]}', ans)))
# ms 一趟处理,直接存贪心结果
def solve():
n, = RI()
a = [0] + RILST()
d = defaultdict(list)
mx = 0
for r in range(1, n + 1):
s = 0
for l in range(r, 0, -1):
s += a[l]
if not d[s] or d[s][-1][1] < l:
d[s].append((l, r))
if len(d[mx]) < len(d[s]):
mx = s
print(len(d[mx]))
print('\n'.join(map(lambda x: f'{x[0]} {x[1]}', d[mx])))
if __name__ == '__main__':
solve()
| py |
1305 | B | B. Kuroni and Simple Stringstime limit per test1 secondmemory limit per test256 megabytesinputstandard inputoutputstandard outputNow that Kuroni has reached 10 years old; he is a big boy and doesn't like arrays of integers as presents anymore. This year he wants a Bracket sequence as a Birthday present. More specifically, he wants a bracket sequence so complex that no matter how hard he tries, he will not be able to remove a simple subsequence!We say that a string formed by nn characters '(' or ')' is simple if its length nn is even and positive, its first n2n2 characters are '(', and its last n2n2 characters are ')'. For example, the strings () and (()) are simple, while the strings )( and ()() are not simple.Kuroni will be given a string formed by characters '(' and ')' (the given string is not necessarily simple). An operation consists of choosing a subsequence of the characters of the string that forms a simple string and removing all the characters of this subsequence from the string. Note that this subsequence doesn't have to be continuous. For example, he can apply the operation to the string ')()(()))', to choose a subsequence of bold characters, as it forms a simple string '(())', delete these bold characters from the string and to get '))()'. Kuroni has to perform the minimum possible number of operations on the string, in such a way that no more operations can be performed on the remaining string. The resulting string does not have to be empty.Since the given string is too large, Kuroni is unable to figure out how to minimize the number of operations. Can you help him do it instead?A sequence of characters aa is a subsequence of a string bb if aa can be obtained from bb by deletion of several (possibly, zero or all) characters.InputThe only line of input contains a string ss (1≤|s|≤10001≤|s|≤1000) formed by characters '(' and ')', where |s||s| is the length of ss.OutputIn the first line, print an integer kk — the minimum number of operations you have to apply. Then, print 2k2k lines describing the operations in the following format:For each operation, print a line containing an integer mm — the number of characters in the subsequence you will remove.Then, print a line containing mm integers 1≤a1<a2<⋯<am1≤a1<a2<⋯<am — the indices of the characters you will remove. All integers must be less than or equal to the length of the current string, and the corresponding subsequence must form a simple string.If there are multiple valid sequences of operations with the smallest kk, you may print any of them.ExamplesInputCopy(()((
OutputCopy1
2
1 3
InputCopy)(
OutputCopy0
InputCopy(()())
OutputCopy1
4
1 2 5 6
NoteIn the first sample; the string is '(()(('. The operation described corresponds to deleting the bolded subsequence. The resulting string is '((('; and no more operations can be performed on it. Another valid answer is choosing indices 22 and 33, which results in the same final string.In the second sample, it is already impossible to perform any operations. | [
"constructive algorithms",
"greedy",
"strings",
"two pointers"
] | def f(x):
y=str(x+1)
return y
s=list(input())
l1=[]
l2=[]
for j in range(len(s)):
if s[j]=="(":
l1.append(j)
else:
l2.append(j)
l2.reverse()
D=[]
c=1
k=0
while c!=0 and len(l1)*len(l2)!=0:
k=k+1
c=0
L=[]
J=[]
for j in range(min(len(l1),len(l2))):
if l1[j]<l2[j]:
c=c+1
L.append(l1[j])
L.append(l2[j])
J.append(j)
if c==0:
k=k-1
ll1=[]
ll2=[]
for j in range(min(len(l1),len(l2))):
if not j in J:
ll1.append(l1[j])
ll2.append(l2[j])
l1=ll1
l2=ll2
L.sort()
L=list(map(f,L))
D.append([c, " ".join(L)])
print(k)
for x in D:
if x[0]!=0:
print(x[0]+x[0])
print(x[1])
| py |
1288 | D | D. Minimax Problemtime limit per test5 secondsmemory limit per test512 megabytesinputstandard inputoutputstandard outputYou are given nn arrays a1a1, a2a2, ..., anan; each array consists of exactly mm integers. We denote the yy-th element of the xx-th array as ax,yax,y.You have to choose two arrays aiai and ajaj (1≤i,j≤n1≤i,j≤n, it is possible that i=ji=j). After that, you will obtain a new array bb consisting of mm integers, such that for every k∈[1,m]k∈[1,m] bk=max(ai,k,aj,k)bk=max(ai,k,aj,k).Your goal is to choose ii and jj so that the value of mink=1mbkmink=1mbk is maximum possible.InputThe first line contains two integers nn and mm (1≤n≤3⋅1051≤n≤3⋅105, 1≤m≤81≤m≤8) — the number of arrays and the number of elements in each array, respectively.Then nn lines follow, the xx-th line contains the array axax represented by mm integers ax,1ax,1, ax,2ax,2, ..., ax,max,m (0≤ax,y≤1090≤ax,y≤109).OutputPrint two integers ii and jj (1≤i,j≤n1≤i,j≤n, it is possible that i=ji=j) — the indices of the two arrays you have to choose so that the value of mink=1mbkmink=1mbk is maximum possible. If there are multiple answers, print any of them.ExampleInputCopy6 5
5 0 3 1 2
1 8 9 1 3
1 2 3 4 5
9 1 0 3 7
2 3 0 6 3
6 4 1 7 0
OutputCopy1 5
| [
"binary search",
"bitmasks",
"dp"
] | import sys
input=lambda:sys.stdin.readline().rstrip()
def trans(a,v):
ret=0
for i in range(len(a)):
ret*=2
ret+=a[i]>=v
return ret
def solve():
n,m=map(int,input().split())
a=[list(map(int,input().split())) for i in range(n)]
bis=[-1,pow(10,9)+1]
ans=[]
while bis[1]-bis[0]>1:
flg=1
mid=sum(bis)//2
dp={}
for i in range(n):
res=trans(a[i],mid)
if not res in dp:
dp[res]=i
for j in dp:
if j|res==pow(2,m)-1:
ans=[dp[j]+1,dp[res]+1]
flg=0
bis[flg]=mid
print(*ans)
T=1
for i in range(T):
solve() | py |
1325 | A | A. EhAb AnD gCdtime limit per test1 secondmemory limit per test256 megabytesinputstandard inputoutputstandard outputYou are given a positive integer xx. Find any such 22 positive integers aa and bb such that GCD(a,b)+LCM(a,b)=xGCD(a,b)+LCM(a,b)=x.As a reminder, GCD(a,b)GCD(a,b) is the greatest integer that divides both aa and bb. Similarly, LCM(a,b)LCM(a,b) is the smallest integer such that both aa and bb divide it.It's guaranteed that the solution always exists. If there are several such pairs (a,b)(a,b), you can output any of them.InputThe first line contains a single integer tt (1≤t≤100)(1≤t≤100) — the number of testcases.Each testcase consists of one line containing a single integer, xx (2≤x≤109)(2≤x≤109).OutputFor each testcase, output a pair of positive integers aa and bb (1≤a,b≤109)1≤a,b≤109) such that GCD(a,b)+LCM(a,b)=xGCD(a,b)+LCM(a,b)=x. It's guaranteed that the solution always exists. If there are several such pairs (a,b)(a,b), you can output any of them.ExampleInputCopy2
2
14
OutputCopy1 1
6 4
NoteIn the first testcase of the sample; GCD(1,1)+LCM(1,1)=1+1=2GCD(1,1)+LCM(1,1)=1+1=2.In the second testcase of the sample, GCD(6,4)+LCM(6,4)=2+12=14GCD(6,4)+LCM(6,4)=2+12=14. | [
"constructive algorithms",
"greedy",
"number theory"
] | import math
from sys import stdin
from collections import Counter, defaultdict, deque
from bisect import bisect_right
from typing import List, DefaultDict
def readarray(typ):
return list(map(typ, stdin.readline().split()))
def readint():
return int(input())
for _ in range(readint()):
x = readint()
print(1, x-1)
| py |
1292 | B | B. Aroma's Searchtime limit per test1 secondmemory limit per test256 megabytesinputstandard inputoutputstandard outputTHE SxPLAY & KIVΛ - 漂流 KIVΛ & Nikki Simmons - PerspectivesWith a new body; our idol Aroma White (or should we call her Kaori Minamiya?) begins to uncover her lost past through the OS space.The space can be considered a 2D plane, with an infinite number of data nodes, indexed from 00, with their coordinates defined as follows: The coordinates of the 00-th node is (x0,y0)(x0,y0) For i>0i>0, the coordinates of ii-th node is (ax⋅xi−1+bx,ay⋅yi−1+by)(ax⋅xi−1+bx,ay⋅yi−1+by) Initially Aroma stands at the point (xs,ys)(xs,ys). She can stay in OS space for at most tt seconds, because after this time she has to warp back to the real world. She doesn't need to return to the entry point (xs,ys)(xs,ys) to warp home.While within the OS space, Aroma can do the following actions: From the point (x,y)(x,y), Aroma can move to one of the following points: (x−1,y)(x−1,y), (x+1,y)(x+1,y), (x,y−1)(x,y−1) or (x,y+1)(x,y+1). This action requires 11 second. If there is a data node at where Aroma is staying, she can collect it. We can assume this action costs 00 seconds. Of course, each data node can be collected at most once. Aroma wants to collect as many data as possible before warping back. Can you help her in calculating the maximum number of data nodes she could collect within tt seconds?InputThe first line contains integers x0x0, y0y0, axax, ayay, bxbx, byby (1≤x0,y0≤10161≤x0,y0≤1016, 2≤ax,ay≤1002≤ax,ay≤100, 0≤bx,by≤10160≤bx,by≤1016), which define the coordinates of the data nodes.The second line contains integers xsxs, ysys, tt (1≤xs,ys,t≤10161≤xs,ys,t≤1016) – the initial Aroma's coordinates and the amount of time available.OutputPrint a single integer — the maximum number of data nodes Aroma can collect within tt seconds.ExamplesInputCopy1 1 2 3 1 0
2 4 20
OutputCopy3InputCopy1 1 2 3 1 0
15 27 26
OutputCopy2InputCopy1 1 2 3 1 0
2 2 1
OutputCopy0NoteIn all three examples; the coordinates of the first 55 data nodes are (1,1)(1,1), (3,3)(3,3), (7,9)(7,9), (15,27)(15,27) and (31,81)(31,81) (remember that nodes are numbered from 00).In the first example, the optimal route to collect 33 nodes is as follows: Go to the coordinates (3,3)(3,3) and collect the 11-st node. This takes |3−2|+|3−4|=2|3−2|+|3−4|=2 seconds. Go to the coordinates (1,1)(1,1) and collect the 00-th node. This takes |1−3|+|1−3|=4|1−3|+|1−3|=4 seconds. Go to the coordinates (7,9)(7,9) and collect the 22-nd node. This takes |7−1|+|9−1|=14|7−1|+|9−1|=14 seconds. In the second example, the optimal route to collect 22 nodes is as follows: Collect the 33-rd node. This requires no seconds. Go to the coordinates (7,9)(7,9) and collect the 22-th node. This takes |15−7|+|27−9|=26|15−7|+|27−9|=26 seconds. In the third example, Aroma can't collect any nodes. She should have taken proper rest instead of rushing into the OS space like that. | [
"brute force",
"constructive algorithms",
"geometry",
"greedy",
"implementation"
] | import sys
input = sys.stdin.buffer.readline
def process(x0, y0, ax, ay, bx, by, xs, ys, t):
a, b = ax, bx
c, d = ay, by
points = []
for i in range(60):
x1 = a**i*x0+((a**i-1)//(a-1))*b
y1 = c**i*y0+((c**i-1)//(c-1))*d
if x1 <= 10**17 and y1 <= 10**17:
points.append([x1, y1])
answer = 0
for earliest in range(len(points)):
x1, y1 = points[earliest]
for latest in range(earliest, len(points)):
x2, y2 = points[latest]
for proximate in range(earliest, latest+1):
x3, y3 = points[proximate]
t1 = abs(x3-xs)+abs(x1-x3)+abs(x2-x1)+abs(y3-ys)+abs(y1-y3)+abs(y2-y1)
t2 = abs(x3-xs)+abs(x2-x3)+abs(x2-x1)+abs(y3-ys)+abs(y2-y3)+abs(y2-y1)
t3 = min(t1, t2)
if t3 <= t:
answer = max(answer, latest-earliest+1)
sys.stdout.write(f'{answer}\n')
return
x0, y0, ax, ay, bx, by = [int(x) for x in input().split()]
xs, ys, t = [int(x) for x in input().split()]
process(x0, y0, ax, ay, bx, by, xs, ys, t) | py |
1303 | B | B. National Projecttime limit per test2 secondsmemory limit per test256 megabytesinputstandard inputoutputstandard outputYour company was appointed to lay new asphalt on the highway of length nn. You know that every day you can either repair one unit of the highway (lay new asphalt over one unit of the highway) or skip repairing.Skipping the repair is necessary because of the climate. The climate in your region is periodical: there are gg days when the weather is good and if you lay new asphalt these days it becomes high-quality pavement; after that, the weather during the next bb days is bad, and if you lay new asphalt these days it becomes low-quality pavement; again gg good days, bb bad days and so on.You can be sure that you start repairing at the start of a good season, in other words, days 1,2,…,g1,2,…,g are good.You don't really care about the quality of the highway, you just want to make sure that at least half of the highway will have high-quality pavement. For example, if the n=5n=5 then at least 33 units of the highway should have high quality; if n=4n=4 then at least 22 units should have high quality.What is the minimum number of days is needed to finish the repair of the whole highway?InputThe first line contains a single integer TT (1≤T≤1041≤T≤104) — the number of test cases.Next TT lines contain test cases — one per line. Each line contains three integers nn, gg and bb (1≤n,g,b≤1091≤n,g,b≤109) — the length of the highway and the number of good and bad days respectively.OutputPrint TT integers — one per test case. For each test case, print the minimum number of days required to repair the whole highway if at least half of it should have high quality.ExampleInputCopy3
5 1 1
8 10 10
1000000 1 1000000
OutputCopy5
8
499999500000
NoteIn the first test case; you can just lay new asphalt each day; since days 1,3,51,3,5 are good.In the second test case, you can also lay new asphalt each day, since days 11-88 are good. | [
"math"
] | for _ in range(int(input())):
n,g,b=map(int,input().split())
if n%2==0:
k=int(n/2)
else:
k=int(n/2)+1
if k%g==0:
x=int(k/g)
else:
x=int(k/g)+1
ans=(x-1)*b+(x-1)*g
gleft=k-(x-1)*g
ans+=gleft
if ans>=n:
print(ans)
else:
print(n) | py |
1285 | B | B. Just Eat It!time limit per test1 secondmemory limit per test256 megabytesinputstandard inputoutputstandard outputToday; Yasser and Adel are at the shop buying cupcakes. There are nn cupcake types, arranged from 11 to nn on the shelf, and there are infinitely many of each type. The tastiness of a cupcake of type ii is an integer aiai. There are both tasty and nasty cupcakes, so the tastiness can be positive, zero or negative.Yasser, of course, wants to try them all, so he will buy exactly one cupcake of each type.On the other hand, Adel will choose some segment [l,r][l,r] (1≤l≤r≤n)(1≤l≤r≤n) that does not include all of cupcakes (he can't choose [l,r]=[1,n][l,r]=[1,n]) and buy exactly one cupcake of each of types l,l+1,…,rl,l+1,…,r.After that they will compare the total tastiness of the cupcakes each of them have bought. Yasser will be happy if the total tastiness of cupcakes he buys is strictly greater than the total tastiness of cupcakes Adel buys regardless of Adel's choice.For example, let the tastinesses of the cupcakes be [7,4,−1][7,4,−1]. Yasser will buy all of them, the total tastiness will be 7+4−1=107+4−1=10. Adel can choose segments [7],[4],[−1],[7,4][7],[4],[−1],[7,4] or [4,−1][4,−1], their total tastinesses are 7,4,−1,117,4,−1,11 and 33, respectively. Adel can choose segment with tastiness 1111, and as 1010 is not strictly greater than 1111, Yasser won't be happy :(Find out if Yasser will be happy after visiting the shop.InputEach test contains multiple test cases. The first line contains the number of test cases tt (1≤t≤1041≤t≤104). The description of the test cases follows.The first line of each test case contains nn (2≤n≤1052≤n≤105).The second line of each test case contains nn integers a1,a2,…,ana1,a2,…,an (−109≤ai≤109−109≤ai≤109), where aiai represents the tastiness of the ii-th type of cupcake.It is guaranteed that the sum of nn over all test cases doesn't exceed 105105.OutputFor each test case, print "YES", if the total tastiness of cupcakes Yasser buys will always be strictly greater than the total tastiness of cupcakes Adel buys regardless of Adel's choice. Otherwise, print "NO".ExampleInputCopy3
4
1 2 3 4
3
7 4 -1
3
5 -5 5
OutputCopyYES
NO
NO
NoteIn the first example; the total tastiness of any segment Adel can choose is less than the total tastiness of all cupcakes.In the second example; Adel will choose the segment [1,2][1,2] with total tastiness 1111, which is not less than the total tastiness of all cupcakes, which is 1010.In the third example, Adel can choose the segment [3,3][3,3] with total tastiness of 55. Note that Yasser's cupcakes' total tastiness is also 55, so in that case, the total tastiness of Yasser's cupcakes isn't strictly greater than the total tastiness of Adel's cupcakes. | [
"dp",
"greedy",
"implementation"
] | def solve():
n = int(input())
aseq = read_ints()
acc = [0]
for ai in aseq:
acc.append(acc[-1] + ai)
# print(acc)
yasser_tastiness = acc[-1]
ans = min(aseq)
for i, e in enumerate(aseq):
if e < 0:
if i + 1 < len(acc):
ans = max(ans, acc[i] - acc[0], acc[-1] - acc[i+1])
else:
ans = max(ans, acc[i] - acc[0])
# print(ans)
return ans < yasser_tastiness
def main():
t = int(input())
output = []
for _ in range(t):
ans = solve()
output.append('YES' if ans else 'NO')
print_lines(output)
def read_ints(): return [int(c) for c in input().split()]
def print_lines(lst): print('\n'.join(map(str, lst)))
if __name__ == "__main__":
from os import environ as env
if 'COMPUTERNAME' in env and 'L2A6HRI' in env['COMPUTERNAME']:
import sys
sys.stdout = open('out.txt', 'w')
sys.stdin = open('in.txt', 'r')
main()
| py |
1311 | A | A. Add Odd or Subtract Eventime limit per test2 secondsmemory limit per test256 megabytesinputstandard inputoutputstandard outputYou are given two positive integers aa and bb.In one move, you can change aa in the following way: Choose any positive odd integer xx (x>0x>0) and replace aa with a+xa+x; choose any positive even integer yy (y>0y>0) and replace aa with a−ya−y. You can perform as many such operations as you want. You can choose the same numbers xx and yy in different moves.Your task is to find the minimum number of moves required to obtain bb from aa. It is guaranteed that you can always obtain bb from aa.You have to answer tt independent test cases.InputThe first line of the input contains one integer tt (1≤t≤1041≤t≤104) — the number of test cases.Then tt test cases follow. Each test case is given as two space-separated integers aa and bb (1≤a,b≤1091≤a,b≤109).OutputFor each test case, print the answer — the minimum number of moves required to obtain bb from aa if you can perform any number of moves described in the problem statement. It is guaranteed that you can always obtain bb from aa.ExampleInputCopy5
2 3
10 10
2 4
7 4
9 3
OutputCopy1
0
2
2
1
NoteIn the first test case; you can just add 11.In the second test case, you don't need to do anything.In the third test case, you can add 11 two times.In the fourth test case, you can subtract 44 and add 11.In the fifth test case, you can just subtract 66. | [
"greedy",
"implementation",
"math"
] | for i in range(int(input())):
a,b = map(int,input().split())
if a == b:
print(0)
else:
print(1 + int((a < b) ^ ((b - a) & 1))) | py |
1311 | D | D. Three Integerstime limit per test2 secondsmemory limit per test256 megabytesinputstandard inputoutputstandard outputYou are given three integers a≤b≤ca≤b≤c.In one move, you can add +1+1 or −1−1 to any of these integers (i.e. increase or decrease any number by one). You can perform such operation any (possibly, zero) number of times, you can even perform this operation several times with one number. Note that you cannot make non-positive numbers using such operations.You have to perform the minimum number of such operations in order to obtain three integers A≤B≤CA≤B≤C such that BB is divisible by AA and CC is divisible by BB.You have to answer tt independent test cases. InputThe first line of the input contains one integer tt (1≤t≤1001≤t≤100) — the number of test cases.The next tt lines describe test cases. Each test case is given on a separate line as three space-separated integers a,ba,b and cc (1≤a≤b≤c≤1041≤a≤b≤c≤104).OutputFor each test case, print the answer. In the first line print resres — the minimum number of operations you have to perform to obtain three integers A≤B≤CA≤B≤C such that BB is divisible by AA and CC is divisible by BB. On the second line print any suitable triple A,BA,B and CC.ExampleInputCopy8
1 2 3
123 321 456
5 10 15
15 18 21
100 100 101
1 22 29
3 19 38
6 30 46
OutputCopy1
1 1 3
102
114 228 456
4
4 8 16
6
18 18 18
1
100 100 100
7
1 22 22
2
1 19 38
8
6 24 48
| [
"brute force",
"math"
] | for _ in range(int(input())):
a,b,c=map(int,input().split())
x=float("inf")
s,d,f=0,0,0
w=[]
for i in range(1,11000):
for j in range(i,11000,i):
for k in range(j,11000,j):
t=abs(a-i)+abs(b-j)+abs(c-k)
if t<x:
x=t
s,d,f=i,j,k
print(x)
print(str(s)+' '+str(d)+" "+str(f)) | py |
1315 | C | C. Restoring Permutationtime limit per test1 secondmemory limit per test256 megabytesinputstandard inputoutputstandard outputYou are given a sequence b1,b2,…,bnb1,b2,…,bn. Find the lexicographically minimal permutation a1,a2,…,a2na1,a2,…,a2n such that bi=min(a2i−1,a2i)bi=min(a2i−1,a2i), or determine that it is impossible.InputEach test contains one or more test cases. The first line contains the number of test cases tt (1≤t≤1001≤t≤100).The first line of each test case consists of one integer nn — the number of elements in the sequence bb (1≤n≤1001≤n≤100).The second line of each test case consists of nn different integers b1,…,bnb1,…,bn — elements of the sequence bb (1≤bi≤2n1≤bi≤2n).It is guaranteed that the sum of nn by all test cases doesn't exceed 100100.OutputFor each test case, if there is no appropriate permutation, print one number −1−1.Otherwise, print 2n2n integers a1,…,a2na1,…,a2n — required lexicographically minimal permutation of numbers from 11 to 2n2n.ExampleInputCopy5
1
1
2
4 1
3
4 1 3
4
2 3 4 5
5
1 5 7 2 8
OutputCopy1 2
-1
4 5 1 2 3 6
-1
1 3 5 6 7 9 2 4 8 10
| [
"greedy"
] | t=int(input())
M=[]
def fonction(n,L):
if max(L)>=2*n:
return -1
elif len(set(L))!=len(L):
return -1
else:
res=list()
D=list()
for k in L:
j=k
ra=[D[i][1] for i in range(len(D))]
while(( j in ra) or (j in L)) and (j<=2*n):
j+=1
D.append((k,j))
res.append(k)
res.append(j)
if sorted(res)==[k for k in range(1,2*n+1)]:
return res
else:
return -1
for i in range(t):
n=int(input())
L=[int(x) for x in input().split()]
M.append(fonction(n,L))
for k in M:
if type(k)==list:
dh=" ".join([str(i) for i in k])
for x in dh.split()[:-1]:
print(int(x) , end =" ")
print(int(dh.split()[-1]))
else:
print(k) | py |
1296 | E1 | E1. String Coloring (easy version)time limit per test1 secondmemory limit per test256 megabytesinputstandard inputoutputstandard outputThis is an easy version of the problem. The actual problems are different; but the easy version is almost a subtask of the hard version. Note that the constraints and the output format are different.You are given a string ss consisting of nn lowercase Latin letters.You have to color all its characters one of the two colors (each character to exactly one color, the same letters can be colored the same or different colors, i.e. you can choose exactly one color for each index in ss).After coloring, you can swap any two neighboring characters of the string that are colored different colors. You can perform such an operation arbitrary (possibly, zero) number of times.The goal is to make the string sorted, i.e. all characters should be in alphabetical order.Your task is to say if it is possible to color the given string so that after coloring it can become sorted by some sequence of swaps. Note that you have to restore only coloring, not the sequence of swaps.InputThe first line of the input contains one integer nn (1≤n≤2001≤n≤200) — the length of ss.The second line of the input contains the string ss consisting of exactly nn lowercase Latin letters.OutputIf it is impossible to color the given string so that after coloring it can become sorted by some sequence of swaps, print "NO" (without quotes) in the first line.Otherwise, print "YES" in the first line and any correct coloring in the second line (the coloring is the string consisting of nn characters, the ii-th character should be '0' if the ii-th character is colored the first color and '1' otherwise).ExamplesInputCopy9
abacbecfd
OutputCopyYES
001010101
InputCopy8
aaabbcbb
OutputCopyYES
01011011
InputCopy7
abcdedc
OutputCopyNO
InputCopy5
abcde
OutputCopyYES
00000
| [
"constructive algorithms",
"dp",
"graphs",
"greedy",
"sortings"
] | n = int(input())
s = input()
ans = "YES"
mx = 'a'
p = ''
lst = 'a'
for i in range(n):
if s[i] >= mx:
mx = s[i]
p += '0'
elif s[i] == mx and i > 0 and s[i - 1] == mx:
p += '0'
elif s[i] == mx or s[i] < mx:
p += '1'
if lst > s[i]:
ans = "NO"
else:
lst = s[i]
if ans == "NO":
print(ans)
else:
print(ans)
print(p)
| py |
1303 | C | C. Perfect Keyboardtime limit per test2 secondsmemory limit per test256 megabytesinputstandard inputoutputstandard outputPolycarp wants to assemble his own keyboard. Layouts with multiple rows are too complicated for him — his keyboard will consist of only one row; where all 2626 lowercase Latin letters will be arranged in some order.Polycarp uses the same password ss on all websites where he is registered (it is bad, but he doesn't care). He wants to assemble a keyboard that will allow to type this password very easily. He doesn't like to move his fingers while typing the password, so, for each pair of adjacent characters in ss, they should be adjacent on the keyboard. For example, if the password is abacaba, then the layout cabdefghi... is perfect, since characters a and c are adjacent on the keyboard, and a and b are adjacent on the keyboard. It is guaranteed that there are no two adjacent equal characters in ss, so, for example, the password cannot be password (two characters s are adjacent).Can you help Polycarp with choosing the perfect layout of the keyboard, if it is possible?InputThe first line contains one integer TT (1≤T≤10001≤T≤1000) — the number of test cases.Then TT lines follow, each containing one string ss (1≤|s|≤2001≤|s|≤200) representing the test case. ss consists of lowercase Latin letters only. There are no two adjacent equal characters in ss.OutputFor each test case, do the following: if it is impossible to assemble a perfect keyboard, print NO (in upper case, it matters in this problem); otherwise, print YES (in upper case), and then a string consisting of 2626 lowercase Latin letters — the perfect layout. Each Latin letter should appear in this string exactly once. If there are multiple answers, print any of them. ExampleInputCopy5
ababa
codedoca
abcda
zxzytyz
abcdefghijklmnopqrstuvwxyza
OutputCopyYES
bacdefghijklmnopqrstuvwxyz
YES
edocabfghijklmnpqrstuvwxyz
NO
YES
xzytabcdefghijklmnopqrsuvw
NO
| [
"dfs and similar",
"greedy",
"implementation"
] | import sys
sys.setrecursionlimit(200000)
import math
from collections import Counter
from collections import defaultdict
from collections import deque
input = sys.stdin.readline
from functools import lru_cache
import heapq
############ ---- Input Functions ---- ############
def inp():
return(int(input()))
def inlt():
return(list(map(int,input().split())))
def insr():
s = input()
return(list(s[:len(s)-1]))
def invr():
return(map(int,input().split()))
def main():
for _ in range(inp()):
password = input().rstrip()
graph = defaultdict(list)
q = deque()
degree = [0 for i in range(26)]
max_degree = 0
for i in range(len(password)):
if i + 1 <len(password) and password[i+1] not in graph[password[i]]:
graph[password[i]].append(password[i+1])
degree[ord(password[i])-ord("a")] = max(degree[ord(password[i])-ord("a")],len(graph[password[i]]))
if i-1 >= 0 and password[i-1] not in graph[password[i]]:
graph[password[i]].append(password[i-1])
degree[ord(password[i])-ord("a")] = max(degree[ord(password[i])-ord("a")],len(graph[password[i]]))
max_degree = max(max_degree,degree[ord(password[i])-ord("a")] )
if max_degree > 2:
print("NO")
continue
s = []
def dfs(element):
s.append(element)
degree[ord(element)-ord("a")] = math.inf
for neigb in graph[element]:
if degree[ord(neigb)-ord("a")] != math.inf :
dfs(neigb)
for i in range(len(degree)):
if degree[i] == 1:
dfs(chr(ord("a")+i))
for i in range(len(degree)):
if degree[i] == 0:
s.append(chr(ord("a")+i))
if len(s) == 26:
print("YES")
print("".join(s))
else:
print("NO")
main() | py |
1315 | B | B. Homecomingtime limit per test1 secondmemory limit per test256 megabytesinputstandard inputoutputstandard outputAfter a long party Petya decided to return home; but he turned out to be at the opposite end of the town from his home. There are nn crossroads in the line in the town, and there is either the bus or the tram station at each crossroad.The crossroads are represented as a string ss of length nn, where si=Asi=A, if there is a bus station at ii-th crossroad, and si=Bsi=B, if there is a tram station at ii-th crossroad. Currently Petya is at the first crossroad (which corresponds to s1s1) and his goal is to get to the last crossroad (which corresponds to snsn).If for two crossroads ii and jj for all crossroads i,i+1,…,j−1i,i+1,…,j−1 there is a bus station, one can pay aa roubles for the bus ticket, and go from ii-th crossroad to the jj-th crossroad by the bus (it is not necessary to have a bus station at the jj-th crossroad). Formally, paying aa roubles Petya can go from ii to jj if st=Ast=A for all i≤t<ji≤t<j. If for two crossroads ii and jj for all crossroads i,i+1,…,j−1i,i+1,…,j−1 there is a tram station, one can pay bb roubles for the tram ticket, and go from ii-th crossroad to the jj-th crossroad by the tram (it is not necessary to have a tram station at the jj-th crossroad). Formally, paying bb roubles Petya can go from ii to jj if st=Bst=B for all i≤t<ji≤t<j.For example, if ss="AABBBAB", a=4a=4 and b=3b=3 then Petya needs: buy one bus ticket to get from 11 to 33, buy one tram ticket to get from 33 to 66, buy one bus ticket to get from 66 to 77. Thus, in total he needs to spend 4+3+4=114+3+4=11 roubles. Please note that the type of the stop at the last crossroad (i.e. the character snsn) does not affect the final expense.Now Petya is at the first crossroad, and he wants to get to the nn-th crossroad. After the party he has left with pp roubles. He's decided to go to some station on foot, and then go to home using only public transport.Help him to choose the closest crossroad ii to go on foot the first, so he has enough money to get from the ii-th crossroad to the nn-th, using only tram and bus tickets.InputEach test contains one or more test cases. The first line contains the number of test cases tt (1≤t≤1041≤t≤104).The first line of each test case consists of three integers a,b,pa,b,p (1≤a,b,p≤1051≤a,b,p≤105) — the cost of bus ticket, the cost of tram ticket and the amount of money Petya has.The second line of each test case consists of one string ss, where si=Asi=A, if there is a bus station at ii-th crossroad, and si=Bsi=B, if there is a tram station at ii-th crossroad (2≤|s|≤1052≤|s|≤105).It is guaranteed, that the sum of the length of strings ss by all test cases in one test doesn't exceed 105105.OutputFor each test case print one number — the minimal index ii of a crossroad Petya should go on foot. The rest of the path (i.e. from ii to nn he should use public transport).ExampleInputCopy5
2 2 1
BB
1 1 1
AB
3 2 8
AABBBBAABB
5 3 4
BBBBB
2 1 1
ABABAB
OutputCopy2
1
3
1
6
| [
"binary search",
"dp",
"greedy",
"strings"
] | import sys
input = lambda: sys.stdin.readline().rstrip()
for _ in range(int(input())):
a, b, p = map(int, input().split())
s = input()
s_len = len(s)
result = 1
f_used = ''
for i in range(s_len - 1, 0, -1):
if s[i - 1] == f_used:
continue
else:
f_used = s[i - 1]
if s[i - 1] == 'A':
p -= a
else:
p -= b
if p < 0:
result = i + 1
break
print(result) | py |
1312 | E | E. Array Shrinkingtime limit per test2 secondsmemory limit per test256 megabytesinputstandard inputoutputstandard outputYou are given an array a1,a2,…,ana1,a2,…,an. You can perform the following operation any number of times: Choose a pair of two neighboring equal elements ai=ai+1ai=ai+1 (if there is at least one such pair). Replace them by one element with value ai+1ai+1. After each such operation, the length of the array will decrease by one (and elements are renumerated accordingly). What is the minimum possible length of the array aa you can get?InputThe first line contains the single integer nn (1≤n≤5001≤n≤500) — the initial length of the array aa.The second line contains nn integers a1,a2,…,ana1,a2,…,an (1≤ai≤10001≤ai≤1000) — the initial array aa.OutputPrint the only integer — the minimum possible length you can get after performing the operation described above any number of times.ExamplesInputCopy5
4 3 2 2 3
OutputCopy2
InputCopy7
3 3 4 4 4 3 3
OutputCopy2
InputCopy3
1 3 5
OutputCopy3
InputCopy1
1000
OutputCopy1
NoteIn the first test; this is one of the optimal sequences of operations: 44 33 22 22 33 →→ 44 33 33 33 →→ 44 44 33 →→ 55 33.In the second test, this is one of the optimal sequences of operations: 33 33 44 44 44 33 33 →→ 44 44 44 44 33 33 →→ 44 44 44 44 44 →→ 55 44 44 44 →→ 55 55 44 →→ 66 44.In the third and fourth tests, you can't perform the operation at all. | [
"dp",
"greedy"
] | from sys import stdin
input = stdin.readline
def solve(i , prev):
if(i == n):return 0
if(dp[i][prev] != -1):return dp[i][prev]
ans , minus = 0 , 1
for j in range(i , n):
if(prev == value[i][j]):
ans = max(ans , solve(j + 1 , prev + 1) + minus)
minus += 1
ans = max(ans , solve(i + 1 , a[i]))
dp[i][prev] = ans
return ans
def answer():
global dp , value
value = [[-1 for i in range(n)] for j in range(n)]
for i in range(n):
value[i][i] = a[i]
for i in range(n - 1 , -1 , -1):
for j in range(i + 1 , n):
for k in range(i , j):
if(value[i][k] == -1 or value[k + 1][j] == -1):continue
if(value[i][k] == value[k + 1][j]):
value[i][j] = value[i][k] + 1
dp = [[-1 for i in range(1501)] for i in range(n)]
ans = solve(0 , 0)
return n - ans
for T in range(1):
n = int(input())
a = list(map(int,input().split()))
print(answer())
| py |
13 | E | E. Holestime limit per test1 secondmemory limit per test64 megabytesinputstdinoutputstdoutLittle Petya likes to play a lot. Most of all he likes to play a game «Holes». This is a game for one person with following rules:There are N holes located in a single row and numbered from left to right with numbers from 1 to N. Each hole has it's own power (hole number i has the power ai). If you throw a ball into hole i it will immediately jump to hole i + ai; then it will jump out of it and so on. If there is no hole with such number, the ball will just jump out of the row. On each of the M moves the player can perform one of two actions: Set the power of the hole a to value b. Throw a ball into the hole a and count the number of jumps of a ball before it jump out of the row and also write down the number of the hole from which it jumped out just before leaving the row. Petya is not good at math, so, as you have already guessed, you are to perform all computations.InputThe first line contains two integers N and M (1 ≤ N ≤ 105, 1 ≤ M ≤ 105) — the number of holes in a row and the number of moves. The second line contains N positive integers not exceeding N — initial values of holes power. The following M lines describe moves made by Petya. Each of these line can be one of the two types: 0 a b 1 a Type 0 means that it is required to set the power of hole a to b, and type 1 means that it is required to throw a ball into the a-th hole. Numbers a and b are positive integers do not exceeding N.OutputFor each move of the type 1 output two space-separated numbers on a separate line — the number of the last hole the ball visited before leaving the row and the number of jumps it made.ExamplesInputCopy8 51 1 1 1 1 2 8 21 10 1 31 10 3 41 2OutputCopy8 78 57 3 | [
"data structures",
"dsu"
] | import sys
input = sys.stdin.readline
n, m = map(int, input().split())
p = list(map(int, input().split()))
size = 320
block = [i // size for i in range(n)]
par, cnt = [0] * n, [0] * n
for i in range(n - 1, -1, -1):
setelah = i + p[i]
if setelah >= n:
par[i] = i + n
cnt[i] = 1
elif block[setelah] == block[i]:
par[i] = par[setelah]
cnt[i] = cnt[setelah] + 1
else:
par[i] = setelah
cnt[i] = 1
ans = []
for _ in range(m):
s = input().strip()
if s[0] == '0':
Z, a, b = s.split()
a = int(a) - 1
b = int(b)
p[a] = b
i = a
while i >= 0 and block[i] == block[a]:
setelah = i + p[i]
if setelah >= n:
par[i] = i + n
cnt[i] = 1
elif block[i] == block[setelah]:
par[i] = par[setelah]
cnt[i] = cnt[setelah] + 1
else:
par[i] = setelah
cnt[i] = 1
i -= 1
else:
Z, a = s.split()
a = int(a) - 1
now = 0
while a < n:
now += cnt[a]
a = par[a]
ans.append(str(a - n + 1) + ' ' + str(now))
print('\n'.join(ans))
| py |
1284 | B | B. New Year and Ascent Sequencetime limit per test2 secondsmemory limit per test1024 megabytesinputstandard inputoutputstandard outputA sequence a=[a1,a2,…,al]a=[a1,a2,…,al] of length ll has an ascent if there exists a pair of indices (i,j)(i,j) such that 1≤i<j≤l1≤i<j≤l and ai<ajai<aj. For example, the sequence [0,2,0,2,0][0,2,0,2,0] has an ascent because of the pair (1,4)(1,4), but the sequence [4,3,3,3,1][4,3,3,3,1] doesn't have an ascent.Let's call a concatenation of sequences pp and qq the sequence that is obtained by writing down sequences pp and qq one right after another without changing the order. For example, the concatenation of the [0,2,0,2,0][0,2,0,2,0] and [4,3,3,3,1][4,3,3,3,1] is the sequence [0,2,0,2,0,4,3,3,3,1][0,2,0,2,0,4,3,3,3,1]. The concatenation of sequences pp and qq is denoted as p+qp+q.Gyeonggeun thinks that sequences with ascents bring luck. Therefore, he wants to make many such sequences for the new year. Gyeonggeun has nn sequences s1,s2,…,sns1,s2,…,sn which may have different lengths. Gyeonggeun will consider all n2n2 pairs of sequences sxsx and sysy (1≤x,y≤n1≤x,y≤n), and will check if its concatenation sx+sysx+sy has an ascent. Note that he may select the same sequence twice, and the order of selection matters.Please count the number of pairs (x,yx,y) of sequences s1,s2,…,sns1,s2,…,sn whose concatenation sx+sysx+sy contains an ascent.InputThe first line contains the number nn (1≤n≤1000001≤n≤100000) denoting the number of sequences.The next nn lines contain the number lili (1≤li1≤li) denoting the length of sisi, followed by lili integers si,1,si,2,…,si,lisi,1,si,2,…,si,li (0≤si,j≤1060≤si,j≤106) denoting the sequence sisi. It is guaranteed that the sum of all lili does not exceed 100000100000.OutputPrint a single integer, the number of pairs of sequences whose concatenation has an ascent.ExamplesInputCopy5
1 1
1 1
1 2
1 4
1 3
OutputCopy9
InputCopy3
4 2 0 2 0
6 9 9 8 8 7 7
1 6
OutputCopy7
InputCopy10
3 62 24 39
1 17
1 99
1 60
1 64
1 30
2 79 29
2 20 73
2 85 37
1 100
OutputCopy72
NoteFor the first example; the following 99 arrays have an ascent: [1,2],[1,2],[1,3],[1,3],[1,4],[1,4],[2,3],[2,4],[3,4][1,2],[1,2],[1,3],[1,3],[1,4],[1,4],[2,3],[2,4],[3,4]. Arrays with the same contents are counted as their occurences. | [
"binary search",
"combinatorics",
"data structures",
"dp",
"implementation",
"sortings"
] | from sys import stdin
input = stdin.readline
#google = lambda : print("Case #%d: "%(T + 1) , end = '')
inp = lambda : list(map(int,input().split()))
def answer():
minv , maxv = [] , []
canPair , ans = n , 0
for x in range(n):
ok , m = False , float('inf')
for i in range(1 , a[x][0] + 1):
m = min(m , a[x][i])
if(m < a[x][i]):
ok = True
break
if(ok):
ans += 2 * canPair - 1
canPair -= 1
else:
minv.append(min(a[x][1:]))
maxv.append(max(a[x][1:]))
j = 0
minv.sort()
maxv.sort()
for i in range(len(minv)):
while(j < len(maxv) and maxv[j] <= minv[i]):
j += 1
ans += len(maxv) - j
return ans
for T in range(1):
n = int(input())
a = [inp() for i in range(n)]
print(answer())
| py |
1301 | C | C. Ayoub's functiontime limit per test1 secondmemory limit per test256 megabytesinputstandard inputoutputstandard outputAyoub thinks that he is a very smart person; so he created a function f(s)f(s), where ss is a binary string (a string which contains only symbols "0" and "1"). The function f(s)f(s) is equal to the number of substrings in the string ss that contains at least one symbol, that is equal to "1".More formally, f(s)f(s) is equal to the number of pairs of integers (l,r)(l,r), such that 1≤l≤r≤|s|1≤l≤r≤|s| (where |s||s| is equal to the length of string ss), such that at least one of the symbols sl,sl+1,…,srsl,sl+1,…,sr is equal to "1". For example, if s=s="01010" then f(s)=12f(s)=12, because there are 1212 such pairs (l,r)(l,r): (1,2),(1,3),(1,4),(1,5),(2,2),(2,3),(2,4),(2,5),(3,4),(3,5),(4,4),(4,5)(1,2),(1,3),(1,4),(1,5),(2,2),(2,3),(2,4),(2,5),(3,4),(3,5),(4,4),(4,5).Ayoub also thinks that he is smarter than Mahmoud so he gave him two integers nn and mm and asked him this problem. For all binary strings ss of length nn which contains exactly mm symbols equal to "1", find the maximum value of f(s)f(s).Mahmoud couldn't solve the problem so he asked you for help. Can you help him? InputThe input consists of multiple test cases. The first line contains a single integer tt (1≤t≤1051≤t≤105) — the number of test cases. The description of the test cases follows.The only line for each test case contains two integers nn, mm (1≤n≤1091≤n≤109, 0≤m≤n0≤m≤n) — the length of the string and the number of symbols equal to "1" in it.OutputFor every test case print one integer number — the maximum value of f(s)f(s) over all strings ss of length nn, which has exactly mm symbols, equal to "1".ExampleInputCopy5
3 1
3 2
3 3
4 0
5 2
OutputCopy4
5
6
0
12
NoteIn the first test case; there exists only 33 strings of length 33, which has exactly 11 symbol, equal to "1". These strings are: s1=s1="100", s2=s2="010", s3=s3="001". The values of ff for them are: f(s1)=3,f(s2)=4,f(s3)=3f(s1)=3,f(s2)=4,f(s3)=3, so the maximum value is 44 and the answer is 44.In the second test case, the string ss with the maximum value is "101".In the third test case, the string ss with the maximum value is "111".In the fourth test case, the only string ss of length 44, which has exactly 00 symbols, equal to "1" is "0000" and the value of ff for that string is 00, so the answer is 00.In the fifth test case, the string ss with the maximum value is "01010" and it is described as an example in the problem statement. | [
"binary search",
"combinatorics",
"greedy",
"math",
"strings"
] | from math import *
from heapq import heappop, heappush
MOD = 10**9 + 7
#input=sys.stdin.readline
#map(int, input().split())
#list(map(int, input().split()))
class Solution():
pass
def areYouCool(n,a,b, amount):
for val in range(1, min(a//amount, n-1) + 1):
if (b//amount) + val >= n:
return True
return False
def solution():
# length n and m fs
# what is the maximam
# 0010010 --> 2 ones --> this means there is only 2
# 5//3 --> 2
# 3 + 3 + 1 --> 7
# 7*4 --> 28
# 000010000001 2
# 000100010000 2
t = int(input())
for _ in range(t):
n, m = map(int, input().split())
if m == 0:
print(0)
continue
zeros = n - m
total = (n*(n+1))//2
if zeros <= m:
print(total - zeros)
continue
m += 1
k = zeros // m
l = zeros % m
# few of them will have k + 1
res = total - l*((k+1)*(k+2))//2 - (m-l)*(k*(k+1))//2
print(res)
def main():
#test()
t = 1
#t = int(input())
for _ in range(t):
solution()
import sys
import threading
sys.setrecursionlimit(10**6)
threading.stack_size(1 << 27)
thread = threading.Thread(target=main)
thread.start(); thread.join()
#main()
| py |
1324 | E | E. Sleeping Scheduletime limit per test2 secondsmemory limit per test256 megabytesinputstandard inputoutputstandard outputVova had a pretty weird sleeping schedule. There are hh hours in a day. Vova will sleep exactly nn times. The ii-th time he will sleep exactly after aiai hours from the time he woke up. You can assume that Vova woke up exactly at the beginning of this story (the initial time is 00). Each time Vova sleeps exactly one day (in other words, hh hours).Vova thinks that the ii-th sleeping time is good if he starts to sleep between hours ll and rr inclusive.Vova can control himself and before the ii-th time can choose between two options: go to sleep after aiai hours or after ai−1ai−1 hours.Your task is to say the maximum number of good sleeping times Vova can obtain if he acts optimally.InputThe first line of the input contains four integers n,h,ln,h,l and rr (1≤n≤2000,3≤h≤2000,0≤l≤r<h1≤n≤2000,3≤h≤2000,0≤l≤r<h) — the number of times Vova goes to sleep, the number of hours in a day and the segment of the good sleeping time.The second line of the input contains nn integers a1,a2,…,ana1,a2,…,an (1≤ai<h1≤ai<h), where aiai is the number of hours after which Vova goes to sleep the ii-th time.OutputPrint one integer — the maximum number of good sleeping times Vova can obtain if he acts optimally.ExampleInputCopy7 24 21 23
16 17 14 20 20 11 22
OutputCopy3
NoteThe maximum number of good times in the example is 33.The story starts from t=0t=0. Then Vova goes to sleep after a1−1a1−1 hours, now the time is 1515. This time is not good. Then Vova goes to sleep after a2−1a2−1 hours, now the time is 15+16=715+16=7. This time is also not good. Then Vova goes to sleep after a3a3 hours, now the time is 7+14=217+14=21. This time is good. Then Vova goes to sleep after a4−1a4−1 hours, now the time is 21+19=1621+19=16. This time is not good. Then Vova goes to sleep after a5a5 hours, now the time is 16+20=1216+20=12. This time is not good. Then Vova goes to sleep after a6a6 hours, now the time is 12+11=2312+11=23. This time is good. Then Vova goes to sleep after a7a7 hours, now the time is 23+22=2123+22=21. This time is also good. | [
"dp",
"implementation"
] | #include <bits/stdc++.h>
#define ll long long
ll a[(int)2e3 + 5], dp[(int)2e3 + 5][(int)2e3 + 5];
ll n, h, l, r;
ll vaild(int x) {
return ((x >= l && x <= r) ? 1ll : 0ll);
}
void solve() {
std::cin >> n >> h >> l >> r;
for(int i = 0; i < n; ++i)
std::cin >> a[i];
for(int i = (n - 1); i >= 0; --i) {
for(int j = h; j >= 0; --j) {
ll v1 = dp[i + 1][(j + a[i]) % h] + vaild((j + a[i]) % h);
ll v2 = dp[i + 1][(((j + a[i] - 1) % h) + h) % h] + vaild((((j + a[i] - 1) % h) + h) % h);
dp[i][j] = std::max(v1, v2);
}
}
std::cout << dp[0][0] << "\n";
}
int main() {
//#
std::ios::sync_with_stdio(false);
std::cin.tie(NULL);
std::cout.tie(NULL);
//#
solve();
//std::cout << "Time : " << 1000 * ((double)clock()) / (double)CLOCKS_PER_SEC << "ms";
return 0;
}
| cpp |
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