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open-r1/verifiable-coding-problems-python-10k | Solve the following coding problem using the programming language python:
There are $X$ people participating in a quiz competition and their IDs have been numbered from $1$ to $X$ (both inclusive). Beth needs to form a team among these $X$ participants. She has been given an integer $Y$. She can choose participants whose ID numbers are divisible by $Y$.
Now that the team is formed, Beth wants to know the strength of her team. The strength of a team is the sum of all the last digits of the team members’ ID numbers.
Can you help Beth in finding the strength of her team?
-----Input:-----
- The first line of the input contains a single integer $T$ denoting the number of test cases. $T$ lines follow
- The first line of each test case contains $X$ and $Y$.
-----Output:-----
- For each test case print the strength of Beth's team
-----Constraints-----
- $1 \leq T \leq 1000$
- $1 \leq X,Y \leq 10^{20}$
-----Sample Input:-----
2
10 3
15 5
-----Sample Output:-----
18
10
-----EXPLANATION:-----
- Example case 1: ID numbers divisible by 3 are 3,6,9 and the sum of the last digits are 3+6+9=18
The input will be stdin and you should print your solution to stdout
Now solve the problem and return the code. | ```python
for _ in range(int(input())):
x, y = map(int, input().split())
ans = 0
for i in range(y, x+1, y):
if i%y == 0:
ans += i%10
print(ans)
``` |
|
open-r1/verifiable-coding-problems-python-10k | Solve the following coding problem using the programming language python:
The Chef has prepared the appetizers in the shapes of letters to spell a special message for the guests. There are n appetizers numbered from 0 to n-1 such that if the appetizers are arrayed in this order, they will display the message. The Chef plans to display them in this order on a table that can be viewed by all guests as they enter. The appetizers will only be served once all guests are seated.
The appetizers are not necessarily finished in the same order as they are numbered. So, when an appetizer is finished the Chef will write the number on a piece of paper and place it beside the appetizer on a counter between the kitchen and the restaurant. A server will retrieve this appetizer and place it in the proper location according to the number written beside it.
The Chef has a penchant for binary numbers. The number of appetizers created is a power of 2, say n = 2k. Furthermore, he has written the number of the appetizer in binary with exactly k bits. That is, binary numbers with fewer than k bits are padded on the left with zeros so they are written with exactly k bits.
Unfortunately, this has unforseen complications. A binary number still "looks" binary when it is written upside down. For example, the binary number "0101" looks like "1010" when read upside down and the binary number "110" looks like "011" (the Chef uses simple vertical lines to denote a 1 bit). The Chef didn't realize that the servers would read the numbers upside down so he doesn't rotate the paper when he places it on the counter. Thus, when the server picks up an appetizer they place it the location indexed by the binary number when it is read upside down.
You are given the message the chef intended to display and you are to display the message that will be displayed after the servers move all appetizers to their locations based on the binary numbers they read.
-----Input-----
The first line consists of a single integer T ≤ 25 indicating the number of test cases to follow. Each test case consists of a single line beginning with an integer 1 ≤ k ≤ 16 followed by a string of precisely 2k characters. The integer and the string are separated by a single space. The string has no spaces and is composed only of lower case letters from `a` to `z`.
-----Output-----
For each test case you are to output the scrambled message on a single line.
-----Example-----
Input:
2
2 chef
4 enjoyourapplepie
Output:
cehf
eayejpuinpopolre
The input will be stdin and you should print your solution to stdout
Now solve the problem and return the code. | ```python
t=int(input())
def reversebinary(bits,n):
bStr=''
for i in range(bits):
if n>0:
bStr=bStr+str(n%2)
else:
bStr=bStr+'0'
n=n>>1
return int(bStr,2)
for i in range(t):
k,msg=input().split()
k=int(k)
newmsg=[]
for j in msg:
newmsg.append(j)
for j in range(len(msg)):
newmsg[reversebinary(k,j)]=msg[j]
print(''.join(newmsg))
``` |
|
open-r1/verifiable-coding-problems-python-10k | Solve the following coding problem using the programming language python:
Polycarp took $n$ videos, the duration of the $i$-th video is $a_i$ seconds. The videos are listed in the chronological order, i.e. the $1$-st video is the earliest, the $2$-nd video is the next, ..., the $n$-th video is the last.
Now Polycarp wants to publish exactly $k$ ($1 \le k \le n$) posts in Instabram. Each video should be a part of a single post. The posts should preserve the chronological order, it means that the first post should contain one or more of the earliest videos, the second post should contain a block (one or more videos) going next and so on. In other words, if the number of videos in the $j$-th post is $s_j$ then:
$s_1+s_2+\dots+s_k=n$ ($s_i>0$), the first post contains the videos: $1, 2, \dots, s_1$; the second post contains the videos: $s_1+1, s_1+2, \dots, s_1+s_2$; the third post contains the videos: $s_1+s_2+1, s_1+s_2+2, \dots, s_1+s_2+s_3$; ... the $k$-th post contains videos: $n-s_k+1,n-s_k+2,\dots,n$.
Polycarp is a perfectionist, he wants the total duration of videos in each post to be the same.
Help Polycarp to find such positive integer values $s_1, s_2, \dots, s_k$ that satisfy all the conditions above.
-----Input-----
The first line contains two integers $n$ and $k$ ($1 \le k \le n \le 10^5$). The next line contains $n$ positive integer numbers $a_1, a_2, \dots, a_n$ ($1 \le a_i \le 10^4$), where $a_i$ is the duration of the $i$-th video.
-----Output-----
If solution exists, print "Yes" in the first line. Print $k$ positive integers $s_1, s_2, \dots, s_k$ ($s_1+s_2+\dots+s_k=n$) in the second line. The total duration of videos in each post should be the same. It can be easily proven that the answer is unique (if it exists).
If there is no solution, print a single line "No".
-----Examples-----
Input
6 3
3 3 1 4 1 6
Output
Yes
2 3 1
Input
3 3
1 1 1
Output
Yes
1 1 1
Input
3 3
1 1 2
Output
No
Input
3 1
1 10 100
Output
Yes
3
The input will be stdin and you should print your solution to stdout
Now solve the problem and return the code. | ```python
import sys
n,k = map(int, sys.stdin.readline().split())
arr = list(map(int, sys.stdin.readline().split()))
tot = sum(arr)
if tot % k != 0:
print('No')
return
tot //= k
idx,cur = 0,0
ans = []
for i in range(n):
cur += arr[i]
idx += 1
if cur == tot:
ans.append(idx)
idx = 0
cur = 0
elif cur > tot:
print('No')
return
if sum(ans) != n:
print('No')
return
print('Yes')
for an in ans:
print(an,end=' ')
``` |
|
open-r1/verifiable-coding-problems-python-10k | Solve the following coding problem using the programming language python:
A string is called bracket sequence if it does not contain any characters other than "(" and ")". A bracket sequence is called regular if it it is possible to obtain correct arithmetic expression by inserting characters "+" and "1" into this sequence. For example, "", "(())" and "()()" are regular bracket sequences; "))" and ")((" are bracket sequences (but not regular ones), and "(a)" and "(1)+(1)" are not bracket sequences at all.
You have a number of strings; each string is a bracket sequence of length $2$. So, overall you have $cnt_1$ strings "((", $cnt_2$ strings "()", $cnt_3$ strings ")(" and $cnt_4$ strings "))". You want to write all these strings in some order, one after another; after that, you will get a long bracket sequence of length $2(cnt_1 + cnt_2 + cnt_3 + cnt_4)$. You wonder: is it possible to choose some order of the strings you have such that you will get a regular bracket sequence? Note that you may not remove any characters or strings, and you may not add anything either.
-----Input-----
The input consists of four lines, $i$-th of them contains one integer $cnt_i$ ($0 \le cnt_i \le 10^9$).
-----Output-----
Print one integer: $1$ if it is possible to form a regular bracket sequence by choosing the correct order of the given strings, $0$ otherwise.
-----Examples-----
Input
3
1
4
3
Output
1
Input
0
0
0
0
Output
1
Input
1
2
3
4
Output
0
-----Note-----
In the first example it is possible to construct a string "(())()(()((()()()())))", which is a regular bracket sequence.
In the second example it is possible to construct a string "", which is a regular bracket sequence.
The input will be stdin and you should print your solution to stdout
Now solve the problem and return the code. | ```python
cnt1 = int(input())
cnt2 = int(input())
cnt3 = int(input())
cnt4 = int(input())
if cnt1 != cnt4:
print(0)
return
if (cnt3 != 0 and cnt1 == 0):
print(0)
return
print(1)
``` |
|
open-r1/verifiable-coding-problems-python-10k | Solve the following coding problem using the programming language python:
Students went into a class to write a test and sat in some way. The teacher thought: "Probably they sat in this order to copy works of each other. I need to rearrange them in such a way that students that were neighbors are not neighbors in a new seating."
The class can be represented as a matrix with n rows and m columns with a student in each cell. Two students are neighbors if cells in which they sit have a common side.
Let's enumerate students from 1 to n·m in order of rows. So a student who initially sits in the cell in row i and column j has a number (i - 1)·m + j. You have to find a matrix with n rows and m columns in which all numbers from 1 to n·m appear exactly once and adjacent numbers in the original matrix are not adjacent in it, or determine that there is no such matrix.
-----Input-----
The only line contains two integers n and m (1 ≤ n, m ≤ 10^5; n·m ≤ 10^5) — the number of rows and the number of columns in the required matrix.
-----Output-----
If there is no such matrix, output "NO" (without quotes).
Otherwise in the first line output "YES" (without quotes), and in the next n lines output m integers which form the required matrix.
-----Examples-----
Input
2 4
Output
YES
5 4 7 2
3 6 1 8
Input
2 1
Output
NO
-----Note-----
In the first test case the matrix initially looks like this:
1 2 3 4
5 6 7 8
It's easy to see that there are no two students that are adjacent in both matrices.
In the second test case there are only two possible seatings and in both of them students with numbers 1 and 2 are neighbors.
The input will be stdin and you should print your solution to stdout
Now solve the problem and return the code. | ```python
n,m=map(int,input().split())
if n==1and m==1:print('YES\n1')
elif n==3and m==3:
print('YES')
print(6, 1, 8)
print(7,5,3)
print(2,9,4)
elif n<4and m<4:print('NO')
elif n==1 or m==1:
t=max(n,m)
a=[i for i in range(2,t+1,2)]
a+=[i for i in range(1,t+1,2)]
print('YES')
for i in a:print(i,end="");print([' ','\n'][m==1],end='')
else:
a=[]
for j in range(n):
a.append([int(i)+int(m*j) for i in range(1,m+1)])
if n<=m:
for j in range(1,m,2):
t=a[0][j]
for i in range(1,n):
a[i-1][j]=a[i][j]
a[n-1][j]=t
for i in range(1,n,2):
r,s=a[i][0],a[i][1]
for j in range(2,m):
a[i][j-2]=a[i][j]
a[i][m-2],a[i][m-1]=r,s
else:
for j in range(1,m,2):
r,s=a[0][j],a[1][j]
for i in range(2,n):
a[i-2][j]=a[i][j]
a[n-2][j], a[n-1][j] = r, s
for i in range(1,n,2):
t=a[i][0]
for j in range(1,m):
a[i][j-1]=a[i][j]
a[i][m-1]=t
print('YES')
for i in range(n):
print(*a[i])
``` |
|
open-r1/verifiable-coding-problems-python-10k | Solve the following coding problem using the programming language python:
A schoolboy named Vasya loves reading books on programming and mathematics. He has recently read an encyclopedia article that described the method of median smoothing (or median filter) and its many applications in science and engineering. Vasya liked the idea of the method very much, and he decided to try it in practice.
Applying the simplest variant of median smoothing to the sequence of numbers a_1, a_2, ..., a_{n} will result a new sequence b_1, b_2, ..., b_{n} obtained by the following algorithm: b_1 = a_1, b_{n} = a_{n}, that is, the first and the last number of the new sequence match the corresponding numbers of the original sequence. For i = 2, ..., n - 1 value b_{i} is equal to the median of three values a_{i} - 1, a_{i} and a_{i} + 1.
The median of a set of three numbers is the number that goes on the second place, when these three numbers are written in the non-decreasing order. For example, the median of the set 5, 1, 2 is number 2, and the median of set 1, 0, 1 is equal to 1.
In order to make the task easier, Vasya decided to apply the method to sequences consisting of zeros and ones only.
Having made the procedure once, Vasya looked at the resulting sequence and thought: what if I apply the algorithm to it once again, and then apply it to the next result, and so on? Vasya tried a couple of examples and found out that after some number of median smoothing algorithm applications the sequence can stop changing. We say that the sequence is stable, if it does not change when the median smoothing is applied to it.
Now Vasya wonders, whether the sequence always eventually becomes stable. He asks you to write a program that, given a sequence of zeros and ones, will determine whether it ever becomes stable. Moreover, if it ever becomes stable, then you should determine what will it look like and how many times one needs to apply the median smoothing algorithm to initial sequence in order to obtain a stable one.
-----Input-----
The first input line of the input contains a single integer n (3 ≤ n ≤ 500 000) — the length of the initial sequence.
The next line contains n integers a_1, a_2, ..., a_{n} (a_{i} = 0 or a_{i} = 1), giving the initial sequence itself.
-----Output-----
If the sequence will never become stable, print a single number - 1.
Otherwise, first print a single integer — the minimum number of times one needs to apply the median smoothing algorithm to the initial sequence before it becomes is stable. In the second line print n numbers separated by a space — the resulting sequence itself.
-----Examples-----
Input
4
0 0 1 1
Output
0
0 0 1 1
Input
5
0 1 0 1 0
Output
2
0 0 0 0 0
-----Note-----
In the second sample the stabilization occurs in two steps: $01010 \rightarrow 00100 \rightarrow 00000$, and the sequence 00000 is obviously stable.
The input will be stdin and you should print your solution to stdout
Now solve the problem and return the code. | ```python
from sys import stdin
#stdin = open('input.txt')
n = int(stdin.readline())
seq = [int(x) for x in stdin.readline().split()]
carry = seq[0]
result = [carry]
mark = False
cur_len = 0
max_len = 0
i = 1
while i < len(seq) - 1:
if mark:
if seq[i] != seq[i + 1]:
cur_len += 1
else:
if cur_len > max_len:
max_len = cur_len
if seq[i] == carry:
result.extend([carry]*cur_len)
else:
result.extend([carry]*(cur_len//2))
result.extend([seq[i]]*(cur_len//2))
result.append(seq[i])
mark = False
cur_len = 0
elif seq[i] != seq[i - 1] and seq[i] != seq[i + 1]:
mark = True
cur_len = 1
carry = seq[i - 1]
else:
result.append(seq[i])
i += 1
if mark:
if cur_len > max_len:
max_len = cur_len
if seq[i] == carry:
result.extend([carry]*cur_len)
else:
result.extend([carry]*(cur_len//2))
result.extend([seq[i]]*(cur_len//2))
result.append(seq[i])
print((max_len + 1)//2)
for x in result:
print(x, end=' ')
``` |
|
open-r1/verifiable-coding-problems-python-10k | Solve the following coding problem using the programming language python:
Chef works as a cook in a restaurant. Each morning, he has to drive down a straight road with length K$K$ to reach the restaurant from his home. Let's describe this road using a coordinate X$X$; the position of Chef's home is X=0$X = 0$ and the position of the restaurant is X=K$X = K$.
The road has exactly two lanes (numbered 1$1$ and 2$2$), but there are N$N$ obstacles (numbered 1$1$ through N$N$) on it. For each valid i$i$, the i$i$-th obstacle blocks the lane Li$L_i$ at the position X=Xi$X = X_i$ and does not block the other lane.
When driving, Chef cannot pass through an obstacle. He can switch lanes in zero time at any integer X$X$-coordinate which does not coincide with the X$X$-coordinate of any obstacle. However, whenever he switches lanes, he cannot switch again until driving for at least D$D$ units of distance, and he can travel only in the direction of increasing X$X$. Chef can start driving in any lane he wants. He can not switch lanes at non-integer X$X$-coordinate.
Sometimes, it is impossible to reach the restaurant without stopping at an obstacle. Find the maximum possible distance Chef can travel before he has to reach an obstacle which is in the same lane as him. If he can avoid all obstacles and reach the restaurant, the answer is K$K$.
-----Input-----
- The first line of the input contains a single integer T$T$ denoting the number of test cases. The description of T$T$ test cases follows.
- The first line of each test case contains three space-separated integers N$N$, K$K$ and D$D$.
- The second line contains N$N$ space-separated integers X1,X2,…,XN$X_1, X_2, \ldots, X_N$.
- The third line contains N$N$ space-separated integers L1,L2,…,LN$L_1, L_2, \ldots, L_N$.
-----Output-----
For each test case, print a single line containing one integer ― the maximum distance Chef can travel.
-----Constraints-----
- 1≤T≤1,000$1 \le T \le 1,000$
- 1≤N≤105$1 \le N \le 10^5$
- 2≤K≤109$2 \le K \le 10^9$
- 1≤D≤109$1 \le D \le 10^9$
- 1≤Xi≤K−1$1 \le X_i \le K-1$ for each valid i$i$
- Xi<Xi+1$X_i < X_{i+1}$ for each valid i$i$
- 1≤Li≤2$1 \le L_i \le 2$ for each valid i$i$
- the sum of N$N$ over all test cases does not exceed 106$10^6$
-----Subtasks-----
Subtask #1 (100 points): original constraints
-----Example Input-----
4
2 10 20
4 7
1 2
4 15 20
4 6 9 13
1 2 2 1
5 10 1
1 3 5 7 9
1 2 1 2 1
2 10 2
4 5
1 2
-----Example Output-----
10
13
10
5
-----Explanation-----
Example case 1: Chef can start in lane 2$2$ and switch to lane 1$1$ at the position X=6$X = 6$, then continue until reaching the restaurant.
Example case 2: Chef can start in lane 2$2$ and switch to lane 1$1$ at X=5$X = 5$. However, he cannot avoid the obstacle at X=13$X = 13$, because he has not driven for at least 20$20$ units since the last switch.
Example case 3: Chef should start in lane 2$2$ and then switch lanes at the positions X=2$X=2$, X=4$X=4$, X=6$X=6$ and X=8$X=8$. This way, he can reach the restaurant.
Example case 4: Chef can start in lane 2$2$ but he cannot escape the second obstacle at X$X$=5 since the first obstacle at X$X$=4 doesn't give enough space for Chef to switch lanes.
The input will be stdin and you should print your solution to stdout
Now solve the problem and return the code. | ```python
# cook your dish here
t = int(input())
for _ in range(t):
n, k, d = list(map(int, input().split()))
x = list(map(int, input().split()))
l = list(map(int, input().split()))
lane = 3 - l[0]
switched = -float('inf')
ans = k
for i in range(n):
if l[i] == lane:
if switched + d < x[i] and x[i - 1] + 1 < x[i]:
lane = 3 - lane
switched = max(x[i - 1] + 1, switched + d)
else:
ans = x[i]
break
print(ans)
``` |
|
open-r1/verifiable-coding-problems-python-10k | Solve the following coding problem using the programming language python:
Greg has an array a = a_1, a_2, ..., a_{n} and m operations. Each operation looks as: l_{i}, r_{i}, d_{i}, (1 ≤ l_{i} ≤ r_{i} ≤ n). To apply operation i to the array means to increase all array elements with numbers l_{i}, l_{i} + 1, ..., r_{i} by value d_{i}.
Greg wrote down k queries on a piece of paper. Each query has the following form: x_{i}, y_{i}, (1 ≤ x_{i} ≤ y_{i} ≤ m). That means that one should apply operations with numbers x_{i}, x_{i} + 1, ..., y_{i} to the array.
Now Greg is wondering, what the array a will be after all the queries are executed. Help Greg.
-----Input-----
The first line contains integers n, m, k (1 ≤ n, m, k ≤ 10^5). The second line contains n integers: a_1, a_2, ..., a_{n} (0 ≤ a_{i} ≤ 10^5) — the initial array.
Next m lines contain operations, the operation number i is written as three integers: l_{i}, r_{i}, d_{i}, (1 ≤ l_{i} ≤ r_{i} ≤ n), (0 ≤ d_{i} ≤ 10^5).
Next k lines contain the queries, the query number i is written as two integers: x_{i}, y_{i}, (1 ≤ x_{i} ≤ y_{i} ≤ m).
The numbers in the lines are separated by single spaces.
-----Output-----
On a single line print n integers a_1, a_2, ..., a_{n} — the array after executing all the queries. Separate the printed numbers by spaces.
Please, do not use the %lld specifier to read or write 64-bit integers in C++. It is preferred to use the cin, cout streams of the %I64d specifier.
-----Examples-----
Input
3 3 3
1 2 3
1 2 1
1 3 2
2 3 4
1 2
1 3
2 3
Output
9 18 17
Input
1 1 1
1
1 1 1
1 1
Output
2
Input
4 3 6
1 2 3 4
1 2 1
2 3 2
3 4 4
1 2
1 3
2 3
1 2
1 3
2 3
Output
5 18 31 20
The input will be stdin and you should print your solution to stdout
Now solve the problem and return the code. | ```python
from sys import stdin, stdout
rd = lambda: list(map(int, stdin.readline().split()))
n, m, k = rd()
a = rd()
b = [rd() for _ in range(m)]
x = [0]*(m+1)
y = [0]*(n+1)
for _ in range(k):
l, r = rd()
x[l-1] += 1
x[r ] -= 1
s = 0
for i in range(m):
l, r, d = b[i]
s += x[i]
y[l-1] += s*d
y[r ] -= s*d
s = 0
for i in range(n):
s += y[i]
a[i] += s
print(' '.join(map(str, a)))
``` |
|
open-r1/verifiable-coding-problems-python-10k | Solve the following coding problem using the programming language python:
Nicholas has an array a that contains n distinct integers from 1 to n. In other words, Nicholas has a permutation of size n.
Nicholas want the minimum element (integer 1) and the maximum element (integer n) to be as far as possible from each other. He wants to perform exactly one swap in order to maximize the distance between the minimum and the maximum elements. The distance between two elements is considered to be equal to the absolute difference between their positions.
-----Input-----
The first line of the input contains a single integer n (2 ≤ n ≤ 100) — the size of the permutation.
The second line of the input contains n distinct integers a_1, a_2, ..., a_{n} (1 ≤ a_{i} ≤ n), where a_{i} is equal to the element at the i-th position.
-----Output-----
Print a single integer — the maximum possible distance between the minimum and the maximum elements Nicholas can achieve by performing exactly one swap.
-----Examples-----
Input
5
4 5 1 3 2
Output
3
Input
7
1 6 5 3 4 7 2
Output
6
Input
6
6 5 4 3 2 1
Output
5
-----Note-----
In the first sample, one may obtain the optimal answer by swapping elements 1 and 2.
In the second sample, the minimum and the maximum elements will be located in the opposite ends of the array if we swap 7 and 2.
In the third sample, the distance between the minimum and the maximum elements is already maximum possible, so we just perform some unnecessary swap, for example, one can swap 5 and 2.
The input will be stdin and you should print your solution to stdout
Now solve the problem and return the code. | ```python
read = lambda: list(map(int, input().split()))
n = int(input())
a = list(read())
x, y = a.index(1), a.index(n)
ans = max(x, y, n - x - 1, n - y - 1)
print(ans)
``` |
|
open-r1/verifiable-coding-problems-python-10k | Solve the following coding problem using the programming language python:
Sereja has two integers — A and B — in 7-ary system. He wants to calculate the number C, such that B * C = A. It is guaranteed that B is a divisor of A.
Please, help Sereja calculate the number C modulo 7L.
-----Input-----
First line of input contains an integer T — the number of test cases. T tests follow.
For each test case, the first line contains the integer A, and the second line contains the integer B, and the third line contains the integer L. A and B are given in 7-ary system.
-----Output-----
Output the answer in 7-ary system.
-----Constraints-----
- 1 ≤ T ≤ 10
- A and B are both positive integers.
- Length of A is a positive integer and doesn't exceed 106.
- L and length of B are positive integers and do not exceed 10000.
-----Subtasks-----
- Sub task #1 (20 points): Length of A is a positive integer and doesn't exceed 20.
- Sub task #2 (30 points): Length of A is a positive integer and doesn't exceed 2000.
- Sub task #3 (50 points): Original constraints.
-----Example-----
Input:3
21
5
10
202
13
1
202
13
2
Output:3
3
13
The input will be stdin and you should print your solution to stdout
Now solve the problem and return the code. | ```python
t=int(input())
while t>0 :
a=int(input())
b=int(input())
l=int(input())
x=0
y=0
z=0
a1=0
b1=0
c1=0
while(a//10!=0 or a%10!=0):
a1+=(a%10+((a//10)%10)*7+((a//100)%10)*49+((a//1000)%10)*343+((a//10000)%10)*2401+((a//100000)%10)*16807+((a//1000000)%10)*117649+((a//10000000)%10)*823543+((a//100000000)%10)*5764801+((a//1000000000)%10)*40353607)*(282475249**x)
x+=1
a//=10000000000
while (b//10!=0 or b%10!=0):
b1+=(b%10+((b//10)%10)*7+((b//100)%10)*49+((b//1000)%10)*343+((b//10000)%10)*2401+((b//100000)%10)*16807+((b//1000000)%10)*117649+((b//10000000)%10)*823543+((b//100000000)%10)*5764801+((b//1000000000)%10)*40353607)*(282475249**y)
y+=1
b//=10000000000
c=(a1//b1)%(7**l)
while z<l:
c1+=(c%7+((c//7)%7)*10+((c//49)%7)*100+((c//343)%7)*1000+((c//2401)%7)*10000+((c//16807)%7)*100000+((c//117649)%7)*1000000+((c//823543)%7)*10000000+((c//5764801)%7)*100000000+((c//40353607)%7)*1000000000)*(10000000000**(z//10))
c//=282475249
z+=10
print(c1)
t-=1
``` |
|
open-r1/verifiable-coding-problems-python-10k | Solve the following coding problem using the programming language python:
Phoenix loves beautiful arrays. An array is beautiful if all its subarrays of length $k$ have the same sum. A subarray of an array is any sequence of consecutive elements.
Phoenix currently has an array $a$ of length $n$. He wants to insert some number of integers, possibly zero, into his array such that it becomes beautiful. The inserted integers must be between $1$ and $n$ inclusive. Integers may be inserted anywhere (even before the first or after the last element), and he is not trying to minimize the number of inserted integers.
-----Input-----
The input consists of multiple test cases. The first line contains an integer $t$ ($1 \le t \le 50$) — the number of test cases.
The first line of each test case contains two integers $n$ and $k$ ($1 \le k \le n \le 100$).
The second line of each test case contains $n$ space-separated integers ($1 \le a_i \le n$) — the array that Phoenix currently has. This array may or may not be already beautiful.
-----Output-----
For each test case, if it is impossible to create a beautiful array, print -1. Otherwise, print two lines.
The first line should contain the length of the beautiful array $m$ ($n \le m \le 10^4$). You don't need to minimize $m$.
The second line should contain $m$ space-separated integers ($1 \le b_i \le n$) — a beautiful array that Phoenix can obtain after inserting some, possibly zero, integers into his array $a$. You may print integers that weren't originally in array $a$.
If there are multiple solutions, print any. It's guaranteed that if we can make array $a$ beautiful, we can always make it with resulting length no more than $10^4$.
-----Example-----
Input
4
4 2
1 2 2 1
4 3
1 2 2 1
3 2
1 2 3
4 4
4 3 4 2
Output
5
1 2 1 2 1
4
1 2 2 1
-1
7
4 3 2 1 4 3 2
-----Note-----
In the first test case, we can make array $a$ beautiful by inserting the integer $1$ at index $3$ (in between the two existing $2$s). Now, all subarrays of length $k=2$ have the same sum $3$. There exists many other possible solutions, for example: $2, 1, 2, 1, 2, 1$ $1, 2, 1, 2, 1, 2$
In the second test case, the array is already beautiful: all subarrays of length $k=3$ have the same sum $5$.
In the third test case, it can be shown that we cannot insert numbers to make array $a$ beautiful.
In the fourth test case, the array $b$ shown is beautiful and all subarrays of length $k=4$ have the same sum $10$. There exist other solutions also.
The input will be stdin and you should print your solution to stdout
Now solve the problem and return the code. | ```python
t = int(input())
for _ in range(t):
n, k = [int(x) for x in input().split()]
a = [int(x) for x in input().split()]
if len(set(a)) > k:
print(-1)
continue
l = list(set(a))
l.extend([1]*(k - len(l)))
print(n*k)
for _ in range(n):
print(*l, end=" ")
print()
``` |
|
open-r1/verifiable-coding-problems-python-10k | Solve the following coding problem using the programming language python:
The statement of this problem is the same as the statement of problem C2. The only difference is that, in problem C1, $n$ is always even, and in C2, $n$ is always odd.
You are given a regular polygon with $2 \cdot n$ vertices (it's convex and has equal sides and equal angles) and all its sides have length $1$. Let's name it as $2n$-gon.
Your task is to find the square of the minimum size such that you can embed $2n$-gon in the square. Embedding $2n$-gon in the square means that you need to place $2n$-gon in the square in such way that each point which lies inside or on a border of $2n$-gon should also lie inside or on a border of the square.
You can rotate $2n$-gon and/or the square.
-----Input-----
The first line contains a single integer $T$ ($1 \le T \le 200$) — the number of test cases.
Next $T$ lines contain descriptions of test cases — one per line. Each line contains single even integer $n$ ($2 \le n \le 200$). Don't forget you need to embed $2n$-gon, not an $n$-gon.
-----Output-----
Print $T$ real numbers — one per test case. For each test case, print the minimum length of a side of the square $2n$-gon can be embedded in. Your answer will be considered correct if its absolute or relative error doesn't exceed $10^{-6}$.
-----Example-----
Input
3
2
4
200
Output
1.000000000
2.414213562
127.321336469
The input will be stdin and you should print your solution to stdout
Now solve the problem and return the code. | ```python
import math
T = int(input())
for _ in range(T):
n = int(input())
print(1/math.tan(math.pi/2/n))
``` |
|
open-r1/verifiable-coding-problems-python-10k | Solve the following coding problem using the programming language python:
Allen dreams of one day owning a enormous fleet of electric cars, the car of the future! He knows that this will give him a big status boost. As Allen is planning out all of the different types of cars he will own and how he will arrange them, he realizes that he has a problem.
Allen's future parking lot can be represented as a rectangle with $4$ rows and $n$ ($n \le 50$) columns of rectangular spaces, each of which can contain at most one car at any time. He imagines having $k$ ($k \le 2n$) cars in the grid, and all the cars are initially in the second and third rows. Each of the cars also has a different designated parking space in the first or fourth row. Allen has to put the cars into corresponding parking places.
[Image] Illustration to the first example.
However, since Allen would never entrust his cars to anyone else, only one car can be moved at a time. He can drive a car from a space in any of the four cardinal directions to a neighboring empty space. Furthermore, Allen can only move one of his cars into a space on the first or fourth rows if it is the car's designated parking space.
Allen knows he will be a very busy man, and will only have time to move cars at most $20000$ times before he realizes that moving cars is not worth his time. Help Allen determine if he should bother parking his cars or leave it to someone less important.
-----Input-----
The first line of the input contains two space-separated integers $n$ and $k$ ($1 \le n \le 50$, $1 \le k \le 2n$), representing the number of columns and the number of cars, respectively.
The next four lines will contain $n$ integers each between $0$ and $k$ inclusive, representing the initial state of the parking lot. The rows are numbered $1$ to $4$ from top to bottom and the columns are numbered $1$ to $n$ from left to right.
In the first and last line, an integer $1 \le x \le k$ represents a parking spot assigned to car $x$ (you can only move this car to this place), while the integer $0$ represents a empty space (you can't move any car to this place).
In the second and third line, an integer $1 \le x \le k$ represents initial position of car $x$, while the integer $0$ represents an empty space (you can move any car to this place).
Each $x$ between $1$ and $k$ appears exactly once in the second and third line, and exactly once in the first and fourth line.
-----Output-----
If there is a sequence of moves that brings all of the cars to their parking spaces, with at most $20000$ car moves, then print $m$, the number of moves, on the first line. On the following $m$ lines, print the moves (one move per line) in the format $i$ $r$ $c$, which corresponds to Allen moving car $i$ to the neighboring space at row $r$ and column $c$.
If it is not possible for Allen to move all the cars to the correct spaces with at most $20000$ car moves, print a single line with the integer $-1$.
-----Examples-----
Input
4 5
1 2 0 4
1 2 0 4
5 0 0 3
0 5 0 3
Output
6
1 1 1
2 1 2
4 1 4
3 4 4
5 3 2
5 4 2
Input
1 2
1
2
1
2
Output
-1
Input
1 2
1
1
2
2
Output
2
1 1 1
2 4 1
-----Note-----
In the first sample test case, all cars are in front of their spots except car $5$, which is in front of the parking spot adjacent. The example shows the shortest possible sequence of moves, but any sequence of length at most $20000$ will be accepted.
In the second sample test case, there is only one column, and the cars are in the wrong order, so no cars can move and the task is impossible.
The input will be stdin and you should print your solution to stdout
Now solve the problem and return the code. | ```python
def main():
n, k = map(int, input().split())
a, b, c, d = (list(map(int, input().split())) for _ in 'abcd')
ss, tt, n2, res = [*b, *c[::-1]], [*a, *d[::-1]], n * 2, []
yx = [*[(2, i + 1) for i in range(n)], *[(3, i) for i in range(n, 0, -1)]]
def park():
for i, s, t, (y, x) in zip(range(n2), ss, tt, yx):
if s == t != 0:
ss[i] = 0
res.append(f'{s} {(1, 4)[y == 3]} {x}')
def rotate():
start = ss.index(0)
for i in range(start - n2, start - 1):
s = ss[i] = ss[i + 1]
if s:
y, x = yx[i]
res.append(f'{s} {y} {x}')
ss[start - 1] = 0
park()
if all(ss):
print(-1)
return
while any(ss):
rotate()
park()
print(len(res), '\n'.join(res), sep='\n')
def __starting_point():
main()
__starting_point()
``` |
|
open-r1/verifiable-coding-problems-python-10k | Solve the following coding problem using the programming language python:
Let $a_1, \ldots, a_n$ be an array of $n$ positive integers. In one operation, you can choose an index $i$ such that $a_i = i$, and remove $a_i$ from the array (after the removal, the remaining parts are concatenated).
The weight of $a$ is defined as the maximum number of elements you can remove.
You must answer $q$ independent queries $(x, y)$: after replacing the $x$ first elements of $a$ and the $y$ last elements of $a$ by $n+1$ (making them impossible to remove), what would be the weight of $a$?
-----Input-----
The first line contains two integers $n$ and $q$ ($1 \le n, q \le 3 \cdot 10^5$) — the length of the array and the number of queries.
The second line contains $n$ integers $a_1$, $a_2$, ..., $a_n$ ($1 \leq a_i \leq n$) — elements of the array.
The $i$-th of the next $q$ lines contains two integers $x$ and $y$ ($x, y \ge 0$ and $x+y < n$).
-----Output-----
Print $q$ lines, $i$-th line should contain a single integer — the answer to the $i$-th query.
-----Examples-----
Input
13 5
2 2 3 9 5 4 6 5 7 8 3 11 13
3 1
0 0
2 4
5 0
0 12
Output
5
11
6
1
0
Input
5 2
1 4 1 2 4
0 0
1 0
Output
2
0
-----Note-----
Explanation of the first query:
After making first $x = 3$ and last $y = 1$ elements impossible to remove, $a$ becomes $[\times, \times, \times, 9, 5, 4, 6, 5, 7, 8, 3, 11, \times]$ (we represent $14$ as $\times$ for clarity).
Here is a strategy that removes $5$ elements (the element removed is colored in red): $[\times, \times, \times, 9, \color{red}{5}, 4, 6, 5, 7, 8, 3, 11, \times]$ $[\times, \times, \times, 9, 4, 6, 5, 7, 8, 3, \color{red}{11}, \times]$ $[\times, \times, \times, 9, 4, \color{red}{6}, 5, 7, 8, 3, \times]$ $[\times, \times, \times, 9, 4, 5, 7, \color{red}{8}, 3, \times]$ $[\times, \times, \times, 9, 4, 5, \color{red}{7}, 3, \times]$ $[\times, \times, \times, 9, 4, 5, 3, \times]$ (final state)
It is impossible to remove more than $5$ elements, hence the weight is $5$.
The input will be stdin and you should print your solution to stdout
Now solve the problem and return the code. | ```python
from sys import stdin
def bitadd(a,w,bit):
x = a
while x <= (len(bit)-1):
bit[x] += w
x += x & (-1 * x)
def bitsum(a,bit):
ret = 0
x = a
while x > 0:
ret += bit[x]
x -= x & (-1 * x)
return ret
class RangeBIT:
def __init__(self,N,indexed):
self.bit1 = [0] * (N+2)
self.bit2 = [0] * (N+2)
self.mode = indexed
def bitadd(self,a,w,bit):
x = a
while x <= (len(bit)-1):
bit[x] += w
x += x & (-1 * x)
def bitsum(self,a,bit):
ret = 0
x = a
while x > 0:
ret += bit[x]
x -= x & (-1 * x)
return ret
def add(self,l,r,w):
l = l + (1-self.mode)
r = r + (1-self.mode)
self.bitadd(l,-1*w*l,self.bit1)
self.bitadd(r,w*r,self.bit1)
self.bitadd(l,w,self.bit2)
self.bitadd(r,-1*w,self.bit2)
def sum(self,l,r):
l = l + (1-self.mode)
r = r + (1-self.mode)
ret = self.bitsum(r,self.bit1) + r * self.bitsum(r,self.bit2)
ret -= self.bitsum(l,self.bit1) + l * self.bitsum(l,self.bit2)
return ret
n,q = list(map(int,stdin.readline().split()))
a = list(map(int,stdin.readline().split()))
qs = [ [] for i in range(n+1) ]
ans = [None] * q
for loop in range(q):
x,y = list(map(int,stdin.readline().split()))
l = x+1
r = n-y
qs[r].append((l,loop))
BIT = [0] * (n+1)
for r in range(1,n+1):
b = r-a[r-1]
if b >= 0:
L = 1
R = r+1
while R-L != 1:
M = (L+R)//2
if bitsum(M,BIT) >= b:
L = M
else:
R = M
if bitsum(L,BIT) >= b:
bitadd(1,1,BIT)
bitadd(L+1,-1,BIT)
for ql,qind in qs[r]:
ans[qind] = bitsum(ql,BIT)
for i in ans:
print (i)
``` |
|
open-r1/verifiable-coding-problems-python-10k | Solve the following coding problem using the programming language python:
Given an array $A_1, A_2, ..., A_N$, count the number of subarrays of array $A$ which are non-decreasing.
A subarray $A[i, j]$, where $1 ≤ i ≤ j ≤ N$ is a sequence of integers $A_i, A_i+1, ..., A_j$.
A subarray $A[i, j]$ is non-decreasing if $A_i ≤ A_i+1 ≤ A_i+2 ≤ ... ≤ A_j$. You have to count the total number of such subarrays.
-----Input-----
-
The first line of input contains an integer $T$ denoting the number of test cases. The description of $T$ test cases follows.
-
The first line of each test case contains a single integer $N$ denoting the size of array.
-
The second line contains $N$ space-separated integers $A_1$, $A_2$, …, $A_N$ denoting the elements of the array.
-----Output-----
For each test case, output in a single line the required answer.
-----Constraints-----
- $1 ≤ T ≤ 5$
- $1 ≤ N ≤ 10^5$
- $1 ≤ A_i ≤ 10^9$
-----Subtasks-----
- Subtask 1 (20 points) : $1 ≤ N ≤ 100$
- Subtask 2 (30 points) : $1 ≤ N ≤ 1000$
- Subtask 3 (50 points) : Original constraints
-----Sample Input:-----
2
4
1 4 2 3
1
5
-----Sample Output:-----
6
1
-----Explanation-----
Example case 1.
All valid subarrays are $A[1, 1], A[1, 2], A[2, 2], A[3, 3], A[3, 4], A[4, 4]$.
Note that singleton subarrays are identically non-decreasing.
Example case 2.
Only single subarray $A[1, 1]$ is non-decreasing.
The input will be stdin and you should print your solution to stdout
Now solve the problem and return the code. | ```python
# cook your dish here
t=int(input())
for i in range(t):
n=int(input())
ar=list(map(int,input().split()))
tot=0
st=0
for j in range(1,n):
if(ar[j-1]>ar[j]):
si=j-st
c=(si*(si+1))//2
tot+=c
st=j
si=n-st
c=(si*(si+1))//2
tot+=c
print(tot)
``` |
|
open-r1/verifiable-coding-problems-python-10k | Solve the following coding problem using the programming language python:
You are given a square matrix $M$ with $N$ rows (numbered $1$ through $N$) and $N$ columns (numbered $1$ through $N$). Initially, all the elements of this matrix are equal to $A$. The matrix is broken down in $N$ steps (numbered $1$ through $N$); note that during this process, some elements of the matrix are simply marked as removed, but all elements are still indexed in the same way as in the original matrix. For each valid $i$, the $i$-th step consists of the following:
- Elements $M_{1, N-i+1}, M_{2, N-i+1}, \ldots, M_{i-1, N-i+1}$ are removed.
- Elements $M_{i, N-i+1}, M_{i, N-i+2}, \ldots, M_{i, N}$ are removed.
- Let's denote the product of all $2i-1$ elements removed in this step by $p_i$. Each of the remaining elements of the matrix (those which have not been removed yet) is multiplied by $p_i$.
Find the sum $p_1 + p_2 + p_3 + \ldots + p_N$. Since this number could be very large, compute it modulo $10^9+7$.
-----Input-----
- The first line of the input contains a single integer $T$ denoting the number of test cases. The description of $T$ test cases follows.
- The first and only line of each test case contains two space-separated integers $N$ and $A$.
-----Output-----
For each test case, print a single line containing one integer ― the sum of products at each step modulo $10^9+7$.
-----Constraints-----
- $1 \le T \le 250$
- $1 \le N \le 10^5$
- $0 \le A \le 10^9$
- the sum of $N$ over all test cases does not exceed $10^5$
-----Example Input-----
1
3 2
-----Example Output-----
511620149
-----Explanation-----
Example case 1:
The input will be stdin and you should print your solution to stdout
Now solve the problem and return the code. | ```python
# cook your dish here
for _ in range(int(input())):
n,k = list(map(int,input().split()))
mod = 10**9+7
s=0
for i in range(1,n+1):
p = pow(k,(2*i)-1,mod)
# print(p)
s=(s+p)%mod
# print(k)
k = (p*k)%mod
print(s)
``` |
|
open-r1/verifiable-coding-problems-python-10k | Solve the following coding problem using the programming language python:
This problem is the most boring one you've ever seen.
Given a sequence of integers a_1, a_2, ..., a_{n} and a non-negative integer h, our goal is to partition the sequence into two subsequences (not necessarily consist of continuous elements). Each element of the original sequence should be contained in exactly one of the result subsequences. Note, that one of the result subsequences can be empty.
Let's define function f(a_{i}, a_{j}) on pairs of distinct elements (that is i ≠ j) in the original sequence. If a_{i} and a_{j} are in the same subsequence in the current partition then f(a_{i}, a_{j}) = a_{i} + a_{j} otherwise f(a_{i}, a_{j}) = a_{i} + a_{j} + h.
Consider all possible values of the function f for some partition. We'll call the goodness of this partiotion the difference between the maximum value of function f and the minimum value of function f.
Your task is to find a partition of the given sequence a that have the minimal possible goodness among all possible partitions.
-----Input-----
The first line of input contains integers n and h (2 ≤ n ≤ 10^5, 0 ≤ h ≤ 10^8). In the second line there is a list of n space-separated integers representing a_1, a_2, ..., a_{n} (0 ≤ a_{i} ≤ 10^8).
-----Output-----
The first line of output should contain the required minimum goodness.
The second line describes the optimal partition. You should print n whitespace-separated integers in the second line. The i-th integer is 1 if a_{i} is in the first subsequence otherwise it should be 2.
If there are several possible correct answers you are allowed to print any of them.
-----Examples-----
Input
3 2
1 2 3
Output
1
1 2 2
Input
5 10
0 1 0 2 1
Output
3
2 2 2 2 2
-----Note-----
In the first sample the values of f are as follows: f(1, 2) = 1 + 2 + 2 = 5, f(1, 3) = 1 + 3 + 2 = 6 and f(2, 3) = 2 + 3 = 5. So the difference between maximum and minimum values of f is 1.
In the second sample the value of h is large, so it's better for one of the sub-sequences to be empty.
The input will be stdin and you should print your solution to stdout
Now solve the problem and return the code. | ```python
n,m=map(int,input().split());a=list(map(int,input().split()));p=0;t=[0]*3
for i in range(n):
if(a[i]<a[p]):p=i
if(n==2):print('0\n1 1\n')
else:
a.sort();t[0]=min(a[0]+a[1]+m,a[1]+a[2]);t[1]=max(a[0]+a[n-1]+m,a[n-2]+a[n-1]);t[2]=(a[n-2]+a[n-1])-(a[0]+a[1])
if(t[1]-t[0]>t[2]):p=n
else:t[2]=t[1]-t[0]
print(t[2])
for i in range(n):print(int(i==p)+1,end=' ')
``` |
|
open-r1/verifiable-coding-problems-python-10k | Solve the following coding problem using the programming language python:
Roger is a robot. He has an arm that is a series of n segments connected to each other. The endpoints of the i-th segment are initially located at points (i - 1, 0) and (i, 0). The endpoint at (i - 1, 0) is colored red and the endpoint at (i, 0) is colored blue for all segments. Thus, the blue endpoint of the i-th segment is touching the red endpoint of the (i + 1)-th segment for all valid i.
Roger can move his arm in two different ways: He can choose some segment and some value. This is denoted as choosing the segment number i and picking some positive l. This change happens as follows: the red endpoint of segment number i and segments from 1 to i - 1 are all fixed in place. Imagine a ray from the red endpoint to the blue endpoint. The blue endpoint and segments i + 1 through n are translated l units in the direction of this ray.
[Image] [Image]
In this picture, the red point labeled A and segments before A stay in place, while the blue point labeled B and segments after B gets translated.
He can choose a segment and rotate it. This is denoted as choosing the segment number i, and an angle a. The red endpoint of the i-th segment will stay fixed in place. The blue endpoint of that segment and segments i + 1 to n will rotate clockwise by an angle of a degrees around the red endpoint.
[Image] [Image]
In this picture, the red point labeled A and segments before A stay in place, while the blue point labeled B and segments after B get rotated around point A.
Roger will move his arm m times. These transformations are a bit complicated, and Roger easily loses track of where the blue endpoint of the last segment is. Help him compute the coordinates of the blue endpoint of the last segment after applying each operation. Note that these operations are cumulative, and Roger's arm may intersect itself arbitrarily during the moves.
-----Input-----
The first line of the input will contain two integers n and m (1 ≤ n, m ≤ 300 000) — the number of segments and the number of operations to perform.
Each of the next m lines contains three integers x_{i}, y_{i} and z_{i} describing a move. If x_{i} = 1, this line describes a move of type 1, where y_{i} denotes the segment number and z_{i} denotes the increase in the length. If x_{i} = 2, this describes a move of type 2, where y_{i} denotes the segment number, and z_{i} denotes the angle in degrees. (1 ≤ x_{i} ≤ 2, 1 ≤ y_{i} ≤ n, 1 ≤ z_{i} ≤ 359)
-----Output-----
Print m lines. The i-th line should contain two real values, denoting the coordinates of the blue endpoint of the last segment after applying operations 1, ..., i. Your answer will be considered correct if its absolute or relative error does not exceed 10^{ - 4}.
Namely, let's assume that your answer for a particular value of a coordinate is a and the answer of the jury is b. The checker program will consider your answer correct if $\frac{|a - b|}{\operatorname{max}(1, b)} \leq 10^{-4}$ for all coordinates.
-----Examples-----
Input
5 4
1 1 3
2 3 90
2 5 48
1 4 1
Output
8.0000000000 0.0000000000
5.0000000000 -3.0000000000
4.2568551745 -2.6691306064
4.2568551745 -3.6691306064
-----Note-----
The following pictures shows the state of the arm after each operation. The coordinates of point F are printed after applying each operation. For simplicity, we only show the blue endpoints of a segment (with the exception for the red endpoint of the first segment). For instance, the point labeled B is the blue endpoint for segment 1 and also the red endpoint for segment 2.
Initial state: [Image] Extend segment 1 by 3. [Image] Rotate segment 3 by 90 degrees clockwise. [Image] Rotate segment 5 by 48 degrees clockwise. [Image] Extend segment 4 by 1. [Image]
The input will be stdin and you should print your solution to stdout
Now solve the problem and return the code. | ```python
from cmath import rect
import sys
import math
from functools import reduce
class SegmentTree():
def __init__(self, L, function = lambda x,y: x+y):
self.function = function
N = self.size = len(L)
M = 1 << N.bit_length()
self.margin = 2*M - N
self.L = [None for i in range(self.margin)] + L
for i in range(M-1, 0, -1):
x, y = self.L[i<<1], self.L[i<<1|1]
self.L[i] = None if x is None or y is None else function(x, y)
def modify(self, pos, value):
p = pos + self.margin
self.L[p] = value
while p > 1:
x, y = self.L[p], self.L[p^1]
if p&1: x, y = y, x
self.L[p>>1] = None if x is None or y is None else self.function(x, y)
p>>=1
def query(self, left, right):
l, r = left + self.margin, right + self.margin
stack = []
void = True
while l < r:
if l&1:
if void:
result = self.L[l]
void = False
else:
result = self.function(result, self.L[l])
l+=1
if r&1:
r-=1
stack.append(self.L[r])
l>>=1
r>>=1
init = stack.pop() if void else result
return reduce(self.function, reversed(stack), init)
def degrect(r, phi):
return rect(r, math.radians(phi))
def vsum(u, v): #u = (x + y*1j, phi)
return (u[0] + v[0]*degrect(1, u[1]), (u[1] + v[1])%360)
def solve(f):
n, m = [int(x) for x in f.readline().split()]
segments = [[1,0] for i in range(n)]
arm = SegmentTree([(1,0) for i in range(n)], vsum)
for line in f:
q, i, a = [int(x) for x in line.split()]
if q == 1:
segments[i-1][0] += a
else:
segments[i-1][1] -= a
arm.modify(i-1, (degrect(segments[i-1][0], segments[i-1][1]), segments[i-1][1]))
query = arm.query(0,n)[0]
print(query.real, query.imag)
solve(sys.stdin)
``` |
|
open-r1/verifiable-coding-problems-python-10k | Solve the following coding problem using the programming language python:
While Grisha was celebrating New Year with Ded Moroz, Misha gifted Sasha a small rectangular pond of size n × m, divided into cells of size 1 × 1, inhabited by tiny evil fishes (no more than one fish per cell, otherwise they'll strife!).
The gift bundle also includes a square scoop of size r × r, designed for fishing. If the lower-left corner of the scoop-net is located at cell (x, y), all fishes inside the square (x, y)...(x + r - 1, y + r - 1) get caught. Note that the scoop-net should lie completely inside the pond when used.
Unfortunately, Sasha is not that skilled in fishing and hence throws the scoop randomly. In order to not frustrate Sasha, Misha decided to release k fishes into the empty pond in such a way that the expected value of the number of caught fishes is as high as possible. Help Misha! In other words, put k fishes in the pond into distinct cells in such a way that when the scoop-net is placed into a random position among (n - r + 1)·(m - r + 1) possible positions, the average number of caught fishes is as high as possible.
-----Input-----
The only line contains four integers n, m, r, k (1 ≤ n, m ≤ 10^5, 1 ≤ r ≤ min(n, m), 1 ≤ k ≤ min(n·m, 10^5)).
-----Output-----
Print a single number — the maximum possible expected number of caught fishes.
You answer is considered correct, is its absolute or relative error does not exceed 10^{ - 9}. Namely, let your answer be a, and the jury's answer be b. Your answer is considered correct, if $\frac{|a - b|}{\operatorname{max}(1,|b|)} \leq 10^{-9}$.
-----Examples-----
Input
3 3 2 3
Output
2.0000000000
Input
12 17 9 40
Output
32.8333333333
-----Note-----
In the first example you can put the fishes in cells (2, 1), (2, 2), (2, 3). In this case, for any of four possible positions of the scoop-net (highlighted with light green), the number of fishes inside is equal to two, and so is the expected value. [Image]
The input will be stdin and you should print your solution to stdout
Now solve the problem and return the code. | ```python
import heapq as hq
from queue import PriorityQueue
import math
n,m,r, k= input().split()
N = int(n)
M = int(m)
R = int(r)
K = int(k)
q = PriorityQueue()
for i in range(1,math.floor((N+1)/2) + 1):
maxi = min(min(i,N-i+1),min(R,N-R+1)) * min(min(R,M-R+1),math.ceil(M/2))
num = M - (2 * min(min(R,M-R+1),math.ceil(M/2))-2)
mult = 2
if(i > math.floor(N/2)):
mult = 1
q.put((-maxi,num * mult,i))
#print(str(maxi) + " " + str(num) + " " + str(mult))
ans = 0
while(K > 0):
pop = q.get()
#print(pop)
a = -1 * pop[0]
b = pop[1]
c = pop[2]
d = min(min(c,N-c+1),min(R,N-R+1))
if(d != a):
# if(q.)
# if(q.get(-(a - d)) != )
mult = 2
if (c > N / 2):
mult = 1
q.put((-(a - d),2*mult,c))
ans += a * min(b,K)
K -= b;
tot = (N-R+1) * (M-R+1)
#print("ANS = " + str(ans))
#print("FINANS = " + str(ans/tot))
print(str(ans/tot))
'''
d = []
for i in range(0,N):
d.append([])
for j in range(0,M):
d[i].append(0)
tot = 0
for i in range(0,N-R+1):
for j in range(0,M-R+1):
for k in range(i,i+R):
for l in range(j,j+R):
d[k][l] += 1
tot += 1
print(N-R+1)*(M-R+1) * (R*R)
print(tot)
print()
for i in d:
print(i)
'''
``` |
|
open-r1/verifiable-coding-problems-python-10k | Solve the following coding problem using the programming language python:
The most dangerous cyborg Jenish is finally entrapped on a narrow bridge over a valley. The bridge is $N$ meters long. For convenience, position $i$ represents a portion of bridge between whose left border is at distance $i-1$ meters from left end and right border is at distance $i$ meters from left end.
There are $N$ batteries placed at positions $1,2...N$. The $i^{th}$ one has energy $a[i]$.
There are two tanks of strength $X$ at positions $0$ and $N+1$. Initially Jenish has $0$ energy.
From now at any second $t$ starting from $1$,
first, Jenish can select any battery placed at position $i$ such that $(t \leq i \leq N-t+1)$ and add $a[i]$ to his energy (every battery can be used atmost once).
Then both the tanks move one meter towards each other. If there are still any batteries present at positions where the tanks are heading, the battery gets destroyed.
At any moment if Jenish's total energy is greater than or equal to $X$, he destroys both the tanks and he escapes the cops. If by the end of $\lfloor \frac {(n+1)}{2}\rfloor^{th}$ second, he can't destroy the tanks, he himself gets destroyed. Find out if he can escape.
-----Input:-----
- The first line consists of a single integer $T$, the number of test cases.
- The first line of each test case contains two space separated integers which represents $N$ and $X$ for that test case respectively.
- The second line of each test case contains $N$ space separated integers, $i^{th}$ of which represents $a[i]$.
-----Output:-----
For each test case, print in a single line, $YES$ if Jenish can escape or $NO$ if he gets destroyed.
-----Constraints:-----
- $1 \leq X \leq 10^9$
- $0 \leq a[i] \leq 10^6$
- $ 1 \leq N \leq 10^5$
- $\Sigma$ $N$ over all the test cases does not exceed $10^5$
-----Sample Input:-----
3
4 8
5 1 4 2
3 4
3 1 2
2 7
5 5
-----Sample Output:-----
YES
YES
NO
-----Explanation-----
For test $1$, in the 1st second, first Jenish will select battery at index $1$ and then the tanks from either side will move one meter closer, thus destroying the battery at index $4$.Then, in the next second, Jenish will select battery at index $3$.Then,tanks will move one meter closer again to destroy the remaining battery at index $2$.But now, Jenish has a total energy of $9$ units which is more than enough to destroy the tanks.
For test $2$, Jenish can use batteries at index $1$ and $2$ to get a total energy of $4$ units.
For test $3$, Jenish can use batteries at index $1$ or $2$ and get a maximum of $5$ units of energy which is less than required.
The input will be stdin and you should print your solution to stdout
Now solve the problem and return the code. | ```python
# cook your dish here
import bisect
for _ in range(int(input())):
n,x=list(map(int,input().split()))
l=list(map(int,input().split()))
battery=[]
power=0
i=0
t=(n+1)//2
while power<x and i<t:
if i==n-i-1:
temp=[-1,l[i]]
else:
temp=sorted([ l[i], l[n-i-1] ])
power+=temp[1]
pos=bisect.bisect_right(battery, temp[1], lo=0, hi=len(battery))
battery.insert(pos,temp[1])
if temp[0]>battery[0]:
power-=battery.pop(0)
power+=temp[0]
pos=bisect.bisect_right(battery, temp[0], lo=0, hi=len(battery))
battery.insert(pos,temp[0])
i+=1
if power>=x:
print('YES')
else:
print('NO')
``` |
|
open-r1/verifiable-coding-problems-python-10k | Solve the following coding problem using the programming language python:
Kabir wants to impress Tara by showing her his problem solving skills. He has decided to give the correct answer to the next question which will be asked by his Algorithms teacher.
The question asked is:
Find the sum of alternate consecutive d$d$ odd numbers from the range L$L$ to R$R$ inclusive.
if d$d$ is 3 and L$L$ is 10 and R$R$ is 34, then the odd numbers between 10 and 34 are 11,13,15,17,19,21,23,25,27,29,31,33$11,13,15,17,19,21,23,25,27,29,31,33$, and the d$d$ alternate odd numbers are 11,13,15,23,25,27$11,13,15,23,25,27$.
You are a friend of Kabir, help him solve the question.
Note:$Note:$ Number of odd number between L$L$ and R$R$ (both inclusive) is a multiple of d$d$.
-----Input:-----
- First line will contain T$T$, number of test cases.
- First line of each test case contains one integer d$d$ .
- Second line of each test case contains two space separated integer L$L$ and R$R$.
-----Output:-----
For each test case, print the sum modulo 1000000007.
-----Constraints:-----
- 1≤T≤106$1 \leq T \leq 10^6$
- 1≤d≤103$1 \leq d \leq 10^3$
- 1≤L<R≤106$1 \leq L < R \leq 10^6$
-----Sample Input:-----
1
3
10 33
-----Sample Output:-----
114
-----EXPLANATION:-----
Sum of alternate odd numbers i.e, 11,13,15,23,25,27 is 114
The input will be stdin and you should print your solution to stdout
Now solve the problem and return the code. | ```python
# cook your dish here
for t in range(int(input().strip())):
d = int(input().strip())
L, R = map(int, input().strip().split(" "))
if L % 2 == 0:
L += 1
sum = (((((R - L + 2)//2)//d)+1)//2) - 1
sum = (sum * 2 * d * (sum + 1) * d) + (sum+1) *d * (L + d -1)
print(sum%1000000007)
``` |
|
open-r1/verifiable-coding-problems-python-10k | Solve the following coding problem using the programming language python:
Nick had received an awesome array of integers $a=[a_1, a_2, \dots, a_n]$ as a gift for his $5$ birthday from his mother. He was already going to explore its various properties but after unpacking he was disappointed a lot because the product $a_1 \cdot a_2 \cdot \dots a_n$ of its elements seemed to him not large enough.
He was ready to throw out the array, but his mother reassured him. She told him, that array would not be spoiled after the following operation: choose any index $i$ ($1 \le i \le n$) and do $a_i := -a_i - 1$.
For example, he can change array $[3, -1, -4, 1]$ to an array $[-4, -1, 3, 1]$ after applying this operation to elements with indices $i=1$ and $i=3$.
Kolya had immediately understood that sometimes it's possible to increase the product of integers of the array a lot. Now he has decided that he wants to get an array with the maximal possible product of integers using only this operation with its elements (possibly zero, one or more times, as many as he wants), it is not forbidden to do this operation several times for the same index.
Help Kolya and print the array with the maximal possible product of elements $a_1 \cdot a_2 \cdot \dots a_n$ which can be received using only this operation in some order.
If there are multiple answers, print any of them.
-----Input-----
The first line contains integer $n$ ($1 \leq n \leq 10^{5}$) — number of integers in the array.
The second line contains $n$ integers $a_1, a_2, \ldots, a_n$ ($-10^{6} \leq a_i \leq 10^{6}$) — elements of the array
-----Output-----
Print $n$ numbers — elements of the array with the maximal possible product of elements which can be received using only this operation in some order from the given array.
If there are multiple answers, print any of them.
-----Examples-----
Input
4
2 2 2 2
Output
-3 -3 -3 -3
Input
1
0
Output
0
Input
3
-3 -3 2
Output
-3 -3 2
The input will be stdin and you should print your solution to stdout
Now solve the problem and return the code. | ```python
n = int(input())
A = list(map(int, input().split()))
if n == 1:
if A[0] >= 0:
print(A[0])
else:
print(-A[0]-1)
return
for i in range(n):
if A[i] < 0:
pass
else:
A[i] = -A[i]-1
if n % 2 == 0:
print(*A)
return
mim = 0
indmim = 0
for i in range(n):
if A[i] < mim:
mim = A[i]
indmim = i
A[indmim] = -A[indmim]-1
print(*A)
``` |
|
open-r1/verifiable-coding-problems-python-10k | Solve the following coding problem using the programming language python:
Chef has a rectangular matrix A of nxm integers. Rows are numbered by integers from 1 to n from top to bottom, columns - from 1 to m from left to right. Ai, j denotes the j-th integer of the i-th row.
Chef wants you to guess his matrix. To guess integers, you can ask Chef questions of next type: "How many integers from submatrix iL, iR, jL, jR are grater than or equal to x and less than or equal to y?". By submatrix iL, iR, jL, jR we mean all elements Ai, j for all iL ≤ i ≤ iR and jL ≤ j ≤ jR.
Also Chef can answer not more than C questions of next type: "What is the sum of integers from submatrix iL, iR, jL, jR?"
As soon as you think you know the Chefs matrix, you can stop asking questions and tell to the Chef your variant of the matrix. Please see "Scoring" part to understand how your solution will be evaluated.
-----Input-----
The first line of the input contains three space-separated integers n, m and C denoting the sizes of the matrix and the maximum number of the second-type questions. After that the judge will answer your questions and evaluate the resuts. Read more about that in the "Interaction with the judge" part of the statement.
-----Interaction with the judge-----
To ask a first-type question you should print to the standard output one line containing seven space-separated integers 1 iL iR jL jR x y. To ask a second-type question you should print one line containing five space-separated integers 2 iL iR jL jR. After that you should read from the standard input one integer - answer to the question. To end the game you should print 3 and starting from next line print n lines, each of them contains m space-separated integers - your variant of the matrix A. After that your program must stop. Remember to flush the output after every line you print.
-----Constraints-----
- 1 ≤ n, m ≤ 2.5 * 105
- 1 ≤ n * m ≤ 2.5 * 105
- 103 ≤ C ≤ 104
- 1 ≤ Ai, j ≤ 50
- 1 ≤ iL ≤ iR ≤ n
- 1 ≤ jL ≤ jR ≤ m
- 1 ≤ x ≤ y ≤ 50
- 0 ≤ number of asked questions ≤ 5 * 105
- 1 ≤ Bi, j ≤ 50
- 1 ≤ a1, a2, a3 ≤ 10
----- Scoring -----
Let B will be the matrix you output and diff = ∑ |Ai, j - Bi, j| for all 1 ≤ i ≤ n, 1 ≤ j ≤ m. The number of questions you asked is questions. The number of integers, you correctly guessed is correct(i. e. the number of elements i, j such that Ai, j = Bi, j).
The score for each test case will be: score = a1 * questions + a2 * diff + a3 * (n * m - correct).
Your goal is to minimize this score.
Your total score for the problem will be the sum of scores on all the test cases.
-----Example-----
Input:
3 3 10
4
0
3
1
6
Output:
1 1 2 1 2 1 3
1 3 3 1 3 1 1
1 3 3 1 3 2 2
1 1 2 3 3 1 1
2 3 3 1 3
3
2 2 1
2 2 1
2 2 2
-----Explanation-----
[1, 2, 3]
A = [3, 2, 1]
[2, 2, 2]
For this test case a1 = 1, a2 = 1 and a3 = 1.
The score for this test case will be 1 * 5 + 1 * 4 + 1 * (9 - 6) = 12.
----- Test data generation -----
There will be four types of the test files.
- Type #1: n = 10, m = 25000
- Type #2: n = 100, m = 2500
- Type #3: n = 250, m = 1000
- Type #4: n = 500, m = 500
There will be 5 test files of each type. During the contest, you will be shown the score for only one test file of each type.
All elements of matrix A are randomly chosen.
For each test case C is randomly chosen from interval [103 .. 104].
For each test case values a1, a2 and a3 are manually chosen.
The input will be stdin and you should print your solution to stdout
Now solve the problem and return the code. | ```python
# CHNGSS.py
import sys
from random import *
n,m,c = list(map(int,input().split()))
arr = [[1]*m for i in range(n)];
saved = 0;
for i in range(n):
for j in range(m):
print(1,(i+1),(i+1),(j+1),(j+1),1,25)
sys.stdout.flush()
a = int(input())
if a == 1 :
saved += 1;
arr[i][j] = randint(1,25);
else:
arr[i][j] = randint(25,50);
print(3);
sys.stdout.flush()
for a in arr :
print(' '.join(map(str,a)));
sys.stdout.flush()
# sys.exit(0);
``` |
|
open-r1/verifiable-coding-problems-python-10k | Solve the following coding problem using the programming language python:
Shubham recently learned the lexicographical order in strings.
Now, he has two strings s1 and s2 of the equal size and Shubham wants to compare those two strings lexicographically.
Help Shubham with the strings comparison.
Note:
Letters are case insensitive.
-----Input-----
First line contains a integer T denoting the number of test cases. Each test case contains two strings of equal size in two separate lines.
-----Output-----
For each test case,
If s1 < s2, print "first".
If s1 > s2, print "second".
If s1=s2, print "equal".
in separate lines.
-----Constraints-----
- 1 ≤ T ≤ 10^2
- 1 ≤ Length of the string ≤ 500
-----Example-----
Input:
2
abc
acb
AB
ba
Output:
first
first
The input will be stdin and you should print your solution to stdout
Now solve the problem and return the code. | ```python
t=eval(input())
while t:
t=t-1
s1=input().lower()
s2=input().lower()
res="equal"
for i in range(len(s1)):
if(s1[i]!=s2[i]):
res="first" if s1[i]<s2[i] else "second"
break
print(res)
``` |
|
open-r1/verifiable-coding-problems-python-10k | Solve the following coding problem using the programming language python:
Chef has a sequence $A_1, A_2, \ldots, A_N$; each element of this sequence is either $0$ or $1$. Appy gave him a string $S$ with length $Q$ describing a sequence of queries. There are two types of queries:
- '!': right-shift the sequence $A$, i.e. replace $A$ by another sequence $B_1, B_2, \ldots, B_N$ satisfying $B_{i+1} = A_i$ for each valid $i$ and $B_1 = A_N$
- '?': find the length of the longest contiguous subsequence of $A$ with length $\le K$ such that each element of this subsequence is equal to $1$
Answer all queries of the second type.
-----Input-----
- The first line of the input contains three space-separated integers $N$, $Q$ and $K$.
- The second line contains $N$ space-separated integers $A_1, A_2, \ldots, A_N$.
- The third line contains a string with length $Q$ describing queries. Each character of this string is either '?', denoting a query of the second type, or '!', denoting a query of the first type.
-----Output-----
For each query of the second type, print a single line containing one integer — the length of the longest required subsequence.
-----Constraints-----
- $1 \le K \le N \le 10^5$
- $1 \le Q \le 3 \cdot 10^5$
- $0 \le A_i \le 1$ for each valid $i$
- $S$ contains only characters '?' and '!'
-----Subtasks-----
Subtask #1 (30 points):
- $1 \le N \le 10^3$
- $1 \le Q \le 3 \cdot 10^3$
Subtask #2 (70 points): original constraints
-----Example Input-----
5 5 3
1 1 0 1 1
?!?!?
-----Example Output-----
2
3
3
-----Explanation-----
- In the first query, there are two longest contiguous subsequences containing only $1$-s: $A_1, A_2$ and $A_4, A_5$. Each has length $2$.
- After the second query, the sequence $A$ is $[1, 1, 1, 0, 1]$.
- In the third query, the longest contiguous subsequence containing only $1$-s is $A_1, A_2, A_3$.
- After the fourth query, $A = [1, 1, 1, 1, 0]$.
- In the fifth query, the longest contiguous subsequence containing only $1$-s is $A_1, A_2, A_3, A_4$ with length $4$. However, we only want subsequences with lengths $\le K$. One of the longest such subsequences is $A_2, A_3, A_4$, with length $3$.
The input will be stdin and you should print your solution to stdout
Now solve the problem and return the code. | ```python
n, q, k = map(int, input().split())
arr = list(map(int, input().split()))
query = list(input())
q_ = len(query)
c1 = query.count('?')
c = arr.count(0)
if c == n:
for i in range(c1):
print(0)
else:
for i in range(q_):
if (i!=0) and (query[i] == '?' and query[i-1] == '?'):
print(max_c)
elif query[i] == '?':
max_c = cnt = 0
for j in range(n):
if (j != n - 1) and (arr[j] == 1 and arr[j + 1] == 1):
cnt += 1
else:
max_c = max(max_c, cnt + 1)
cnt = 0
if k < max_c:
max_c = k
break
print(max_c)
elif query[i] == '!':
temp = arr[n - 1]
del arr[n - 1]
arr.insert(0, temp)
``` |
|
open-r1/verifiable-coding-problems-python-10k | Solve the following coding problem using the programming language python:
Niwel is a little golden bear. As everyone knows, bears live in forests, but Niwel got tired of seeing all the trees so he decided to move to the city.
In the city, Niwel took on a job managing bears to deliver goods. The city that he lives in can be represented as a directed graph with n nodes and m edges. Each edge has a weight capacity. A delivery consists of a bear carrying weights with their bear hands on a simple path from node 1 to node n. The total weight that travels across a particular edge must not exceed the weight capacity of that edge.
Niwel has exactly x bears. In the interest of fairness, no bear can rest, and the weight that each bear carries must be exactly the same. However, each bear may take different paths if they like.
Niwel would like to determine, what is the maximum amount of weight he can deliver (it's the sum of weights carried by bears). Find the maximum weight.
-----Input-----
The first line contains three integers n, m and x (2 ≤ n ≤ 50, 1 ≤ m ≤ 500, 1 ≤ x ≤ 100 000) — the number of nodes, the number of directed edges and the number of bears, respectively.
Each of the following m lines contains three integers a_{i}, b_{i} and c_{i} (1 ≤ a_{i}, b_{i} ≤ n, a_{i} ≠ b_{i}, 1 ≤ c_{i} ≤ 1 000 000). This represents a directed edge from node a_{i} to b_{i} with weight capacity c_{i}. There are no self loops and no multiple edges from one city to the other city. More formally, for each i and j that i ≠ j it's guaranteed that a_{i} ≠ a_{j} or b_{i} ≠ b_{j}. It is also guaranteed that there is at least one path from node 1 to node n.
-----Output-----
Print one real value on a single line — the maximum amount of weight Niwel can deliver if he uses exactly x bears. Your answer will be considered correct if its absolute or relative error does not exceed 10^{ - 6}.
Namely: let's assume that your answer is a, and the answer of the jury is b. The checker program will consider your answer correct if $\frac{|a - b|}{\operatorname{max}(1, b)} \leq 10^{-6}$.
-----Examples-----
Input
4 4 3
1 2 2
2 4 1
1 3 1
3 4 2
Output
1.5000000000
Input
5 11 23
1 2 3
2 3 4
3 4 5
4 5 6
1 3 4
2 4 5
3 5 6
1 4 2
2 5 3
1 5 2
3 2 30
Output
10.2222222222
-----Note-----
In the first sample, Niwel has three bears. Two bears can choose the path $1 \rightarrow 3 \rightarrow 4$, while one bear can choose the path $1 \rightarrow 2 \rightarrow 4$. Even though the bear that goes on the path $1 \rightarrow 2 \rightarrow 4$ can carry one unit of weight, in the interest of fairness, he is restricted to carry 0.5 units of weight. Thus, the total weight is 1.5 units overall. Note that even though Niwel can deliver more weight with just 2 bears, he must use exactly 3 bears on this day.
The input will be stdin and you should print your solution to stdout
Now solve the problem and return the code. | ```python
from collections import defaultdict, deque
adj = defaultdict(lambda: defaultdict(lambda: 0))
def bfs(graph, inicio, destino, parent):
parent.clear()
queue = deque()
queue.append([inicio, float("Inf")])
parent[inicio] = -2
while (len(queue)):
current, flow = queue.popleft()
for i in adj[current]:
if parent[i] == -1 and graph[current][i] > 0:
parent[i] = current
flow = min(flow, graph[current][i])
if i == destino:
return flow
queue.append((i, flow))
return 0
def maxflow(graph, inicio, destino):
flow = 0
parent = defaultdict(lambda: -1)
while True:
t = bfs(graph, inicio, destino, parent)
if t:
flow += t
current = destino
while current != inicio:
prev = parent[current]
graph[prev][current] -= t
graph[current][prev] += t
current = prev
else:
break
return flow
n, m, x = [int(i) for i in input().split()]
for _ in range(m):
t = [int(i) for i in input().split()]
adj[t[0]][t[1]] = t[2]
def check(k):
meh = defaultdict(lambda: defaultdict(lambda: 0))
for i in adj:
for j in adj[i]:
ww = adj[i][j] // k
meh[i][j] = ww
flow = maxflow(meh, 1, n)
return flow
lo = 1 / x
hi = check(1)
for _ in range(70):
mid = (hi + lo) / 2
if hi-lo<0.0000000001:
break
if check(mid)>=x:
lo = mid
else:
hi = mid
print(format(lo * x, '.9f'))
``` |
|
open-r1/verifiable-coding-problems-python-10k | Solve the following coding problem using the programming language python:
Vanya wants to minimize a tree. He can perform the following operation multiple times: choose a vertex v, and two disjoint (except for v) paths of equal length a_0 = v, a_1, ..., a_{k}, and b_0 = v, b_1, ..., b_{k}. Additionally, vertices a_1, ..., a_{k}, b_1, ..., b_{k} must not have any neighbours in the tree other than adjacent vertices of corresponding paths. After that, one of the paths may be merged into the other, that is, the vertices b_1, ..., b_{k} can be effectively erased: [Image]
Help Vanya determine if it possible to make the tree into a path via a sequence of described operations, and if the answer is positive, also determine the shortest length of such path.
-----Input-----
The first line of input contains the number of vertices n (2 ≤ n ≤ 2·10^5).
Next n - 1 lines describe edges of the tree. Each of these lines contains two space-separated integers u and v (1 ≤ u, v ≤ n, u ≠ v) — indices of endpoints of the corresponding edge. It is guaranteed that the given graph is a tree.
-----Output-----
If it is impossible to obtain a path, print -1. Otherwise, print the minimum number of edges in a possible path.
-----Examples-----
Input
6
1 2
2 3
2 4
4 5
1 6
Output
3
Input
7
1 2
1 3
3 4
1 5
5 6
6 7
Output
-1
-----Note-----
In the first sample case, a path of three edges is obtained after merging paths 2 - 1 - 6 and 2 - 4 - 5.
It is impossible to perform any operation in the second sample case. For example, it is impossible to merge paths 1 - 3 - 4 and 1 - 5 - 6, since vertex 6 additionally has a neighbour 7 that is not present in the corresponding path.
The input will be stdin and you should print your solution to stdout
Now solve the problem and return the code. | ```python
import math,string,itertools,fractions,heapq,collections,re,array,bisect,sys,random,time
sys.setrecursionlimit(10**7)
inf = 10**20
mod = 10**9 + 7
def LI(): return list(map(int, input().split()))
def II(): return int(input())
def LS(): return input().split()
def S(): return input()
def main():
n = II()
d = collections.defaultdict(set)
for _ in range(n-1):
a,b = LI()
d[a].add(b)
d[b].add(a)
memo = [-1] * (n+1)
def path(t,s):
ps = set()
dt = list(d[t])
for k in dt:
if memo[k] < 0:
continue
ps.add(memo[k])
if s == -1 and len(ps) == 2:
memo[t] = sum(ps) + 2
return memo[t]
if len(ps) > 1:
return -t
if len(ps) == 0:
memo[t] = 0
return 0
memo[t] = list(ps)[0] + 1
return memo[t]
def _path(tt,ss):
f = [False] * (n+1)
q = [(tt,ss)]
tq = []
qi = 0
while len(q) > qi:
t,s = q[qi]
for k in d[t]:
if k == s or memo[k] >= 0:
continue
q.append((k,t))
qi += 1
for t,s in q[::-1]:
r = path(t,s)
if r < 0:
return r
return memo[tt]
t = _path(1,-1)
if t < 0:
t = _path(-t,-1)
if t > 0:
while t%2 == 0:
t//=2
return t
return -1
print(main())
``` |
|
open-r1/verifiable-coding-problems-python-10k | Solve the following coding problem using the programming language python:
On the Literature lesson Sergei noticed an awful injustice, it seems that some students are asked more often than others.
Seating in the class looks like a rectangle, where n rows with m pupils in each.
The teacher asks pupils in the following order: at first, she asks all pupils from the first row in the order of their seating, then she continues to ask pupils from the next row. If the teacher asked the last row, then the direction of the poll changes, it means that she asks the previous row. The order of asking the rows looks as follows: the 1-st row, the 2-nd row, ..., the n - 1-st row, the n-th row, the n - 1-st row, ..., the 2-nd row, the 1-st row, the 2-nd row, ...
The order of asking of pupils on the same row is always the same: the 1-st pupil, the 2-nd pupil, ..., the m-th pupil.
During the lesson the teacher managed to ask exactly k questions from pupils in order described above. Sergei seats on the x-th row, on the y-th place in the row. Sergei decided to prove to the teacher that pupils are asked irregularly, help him count three values: the maximum number of questions a particular pupil is asked, the minimum number of questions a particular pupil is asked, how many times the teacher asked Sergei.
If there is only one row in the class, then the teacher always asks children from this row.
-----Input-----
The first and the only line contains five integers n, m, k, x and y (1 ≤ n, m ≤ 100, 1 ≤ k ≤ 10^18, 1 ≤ x ≤ n, 1 ≤ y ≤ m).
-----Output-----
Print three integers: the maximum number of questions a particular pupil is asked, the minimum number of questions a particular pupil is asked, how many times the teacher asked Sergei.
-----Examples-----
Input
1 3 8 1 1
Output
3 2 3
Input
4 2 9 4 2
Output
2 1 1
Input
5 5 25 4 3
Output
1 1 1
Input
100 100 1000000000000000000 100 100
Output
101010101010101 50505050505051 50505050505051
-----Note-----
The order of asking pupils in the first test: the pupil from the first row who seats at the first table, it means it is Sergei; the pupil from the first row who seats at the second table; the pupil from the first row who seats at the third table; the pupil from the first row who seats at the first table, it means it is Sergei; the pupil from the first row who seats at the second table; the pupil from the first row who seats at the third table; the pupil from the first row who seats at the first table, it means it is Sergei; the pupil from the first row who seats at the second table;
The order of asking pupils in the second test: the pupil from the first row who seats at the first table; the pupil from the first row who seats at the second table; the pupil from the second row who seats at the first table; the pupil from the second row who seats at the second table; the pupil from the third row who seats at the first table; the pupil from the third row who seats at the second table; the pupil from the fourth row who seats at the first table; the pupil from the fourth row who seats at the second table, it means it is Sergei; the pupil from the third row who seats at the first table;
The input will be stdin and you should print your solution to stdout
Now solve the problem and return the code. | ```python
n, m, k, x, y = list(map(int, input().split()))
ans = [[0] * m for x in range(n)]
onebig = (2*n-2)*m or m
oo = k // onebig
for i in range(n):
for j in range(m):
if i == 0 or i == n-1:
ans[i][j] += oo
k -= oo
else:
ans[i][j] += 2*oo
k -= 2*oo
from itertools import chain
for i in chain(list(range(n)), list(range(n-2, 0, -1))):
if not k:
break
for j in range(m):
if not k:
break
ans[i][j] += 1
k -= 1
_max = max(list(map(max, ans)))
_min = min(list(map(min, ans)))
_ans = ans[x-1][y-1]
print(_max, _min, _ans)
``` |
|
open-r1/verifiable-coding-problems-python-10k | Solve the following coding problem using the programming language python:
As Will is stuck in the Upside Down, he can still communicate with his mom, Joyce, through the Christmas lights (he can turn them on and off with his mind). He can't directly tell his mom where he is, because the monster that took him to the Upside Down will know and relocate him. [Image]
Thus, he came up with a puzzle to tell his mom his coordinates. His coordinates are the answer to the following problem.
A string consisting only of parentheses ('(' and ')') is called a bracket sequence. Some bracket sequence are called correct bracket sequences. More formally: Empty string is a correct bracket sequence. if s is a correct bracket sequence, then (s) is also a correct bracket sequence. if s and t are correct bracket sequences, then st (concatenation of s and t) is also a correct bracket sequence.
A string consisting of parentheses and question marks ('?') is called pretty if and only if there's a way to replace each question mark with either '(' or ')' such that the resulting string is a non-empty correct bracket sequence.
Will gave his mom a string s consisting of parentheses and question marks (using Morse code through the lights) and his coordinates are the number of pairs of integers (l, r) such that 1 ≤ l ≤ r ≤ |s| and the string s_{l}s_{l} + 1... s_{r} is pretty, where s_{i} is i-th character of s.
Joyce doesn't know anything about bracket sequences, so she asked for your help.
-----Input-----
The first and only line of input contains string s, consisting only of characters '(', ')' and '?' (2 ≤ |s| ≤ 5000).
-----Output-----
Print the answer to Will's puzzle in the first and only line of output.
-----Examples-----
Input
((?))
Output
4
Input
??()??
Output
7
-----Note-----
For the first sample testcase, the pretty substrings of s are: "(?" which can be transformed to "()". "?)" which can be transformed to "()". "((?)" which can be transformed to "(())". "(?))" which can be transformed to "(())".
For the second sample testcase, the pretty substrings of s are: "??" which can be transformed to "()". "()". "??()" which can be transformed to "()()". "?()?" which can be transformed to "(())". "??" which can be transformed to "()". "()??" which can be transformed to "()()". "??()??" which can be transformed to "()()()".
The input will be stdin and you should print your solution to stdout
Now solve the problem and return the code. | ```python
def main():
s = input()
l = len(s)
pretty_count = 0
for i in range(l):
left_paren_count = 0
right_paren_count = 0
wild_count = 0
for j in range(i, l):
if s[j] == '(':
left_paren_count += 1
elif s[j] == ')':
right_paren_count += 1
else:
wild_count += 1
if left_paren_count + wild_count < right_paren_count:
break
if left_paren_count < wild_count + right_paren_count:
# Should fix one '?' as '('
wild_count -= 1
left_paren_count += 1
if wild_count < 0:
break
if left_paren_count == wild_count + right_paren_count:
pretty_count += 1
print(pretty_count)
def __starting_point():
main()
__starting_point()
``` |
|
open-r1/verifiable-coding-problems-python-10k | Solve the following coding problem using the programming language python:
You are given a binary string S. You need to transform this string into another string of equal length consisting only of zeros, with the minimum number of operations.
A single operation consists of taking some prefix of the string S and flipping all its values. That is, change all the 0s in this prefix to 1s, and all the 1s in the prefix to 0s. You can use this operation as many number of times as you want over any prefix of the string.
-----Input-----
The only line of the input contains the binary string, S .
-----Output-----
Output a single line containing one integer, the minimum number of operations that are needed to transform the given string S into the string of equal length consisting only of zeros.
-----Constraints-----
- 1 ≤ |S| ≤ 100,000
-----Subtasks-----
- Subtask #1 (30 points): 1 ≤ |S| ≤ 2000
- Subtask #2 (70 points): Original constraints.
-----Example-----
Input:
01001001
Output:
6
-----Explanation-----
For the given sample case, let us look at the way where we achieved minimum number of operations.
Operation 1: You flip values in the prefix of length 8 and transform the string into 10110110
Operation 2: You flip values in the prefix of length 7 and transform the string into 01001000
Operation 3: You flip values in the prefix of length 5 and transform the string into 10110000
Operation 4: You flip values in the prefix of length 4 and transform the string into 01000000
Operation 5: You flip values in the prefix of length 2 and transform the string into 10000000
Operation 6: You flip values in the prefix of length 1 and finally, transform the string into 00000000
The input will be stdin and you should print your solution to stdout
Now solve the problem and return the code. | ```python
# cook your dish here
s=input()
s1=s[::-1]
arr=[]
cnt=0
for i in range(len(s1)):
arr.append(s1[i])
for i in range(len(arr)):
if(arr[i]=="1"):
for j in range(i,len(arr)):
if(arr[j]=="1"):
arr[j]="0"
else:
arr[j]="1"
cnt+=1
print(cnt)
``` |
|
open-r1/verifiable-coding-problems-python-10k | Solve the following coding problem using the programming language python:
It's year 2018 and it's Christmas time! Before going for vacations, students of Hogwarts School of Witchcraft and Wizardry had their end semester exams.
$N$ students attended the semester exam. Once the exam was over, their results were displayed as either "Pass" or "Fail" behind their magic jacket which they wore. A student cannot see his/her result but can see everyone else's results. Each of $N$ students count the number of passed students they can see.
Given the number of "Pass" verdicts that each of the $N$ students counted, we have to figure out conclusively, the number of students who failed, or report that there is some inconsistency or that we cannot be sure.
-----Input:-----
- First line will contain $T$, number of testcases. Then the testcases follow.
- The first line of each test case will contain $N$, representing the number of students who attended the exam.
- Next line contains $N$ spaced integers representing the number of "Pass" counted by each of the $N$ students.
-----Output:-----
- For each test case, output the answer in a single line.
- If the counts reported by the students are not consistent with each other or if it's not possible to predict the number of failed students from the given input, then print -1.
-----Constraints-----
- $1 \leq T \leq 50$
- $1 \leq N \leq 10^{5}$
- $0 \leq$ Count given by each Student $\leq 10^{5}$
-----Sample Input:-----
1
4
3 2 2 2
-----Sample Output:-----
1
-----EXPLANATION:-----
There are 4 students, and they counted the number of passed students as 3,2,2,2. The first student can see that all others have passed, and all other students can see only 2 students who have passed. Hence, the first student must have failed, and others have all passed. Hence, the answer is 1.
The input will be stdin and you should print your solution to stdout
Now solve the problem and return the code. | ```python
for _ in range(int(input())):
n = int(input())
a = list(map(int, input().split()))
s = set(a)
if n == 1 or len(s) > 2:
print(-1)
continue
if len(s) == 1:
x = s.pop()
if x == 0:
print(n)
elif x == n-1:
print(0)
else:
print(-1)
continue
x, y = sorted(s)
xc, yc = a.count(x), a.count(y)
if xc == y and xc == x + 1:
print(yc)
else:
print(-1)
``` |
|
open-r1/verifiable-coding-problems-python-10k | Solve the following coding problem using the programming language python:
Chef has a nice complete binary tree in his garden. Complete means that each node has exactly two sons, so the tree is infinite. Yesterday he had enumerated the nodes of the tree in such a way:
- Let's call the nodes' level a number of nodes that occur on the way to this node from the root, including this node. This way, only the root has the level equal to 1, while only its two sons has the level equal to 2.
- Then, let's take all the nodes with the odd level and enumerate them with consecutive odd numbers, starting from the smallest levels and the leftmost nodes, going to the rightmost nodes and the highest levels.
- Then, let's take all the nodes with the even level and enumerate them with consecutive even numbers, starting from the smallest levels and the leftmost nodes, going to the rightmost nodes and the highest levels.
- For the better understanding there is an example:
1
/ \
2 4
/ \ / \
3 5 7 9
/ \ / \ / \ / \
6 8 10 12 14 16 18 20
Here you can see the visualization of the process. For example, in odd levels, the root was enumerated first, then, there were enumerated roots' left sons' sons and roots' right sons' sons.
You are given the string of symbols, let's call it S. Each symbol is either l or r. Naturally, this sequence denotes some path from the root, where l means going to the left son and r means going to the right son.
Please, help Chef to determine the number of the last node in this path.
-----Input-----
The first line contains single integer T number of test cases.
Each of next T lines contain a string S consisting only of the symbols l and r.
-----Output-----
Per each line output the number of the last node in the path, described by S, modulo 109+7.
-----Constraints-----
- 1 ≤ |T| ≤ 5
- 1 ≤ |S| ≤ 10^5
- Remember that the tree is infinite, so each path described by appropriate S is a correct one.
-----Example-----
Input:
4
lrl
rll
r
lllr
Output:
10
14
4
13
-----Explanation-----
See the example in the statement for better understanding the samples.
The input will be stdin and you should print your solution to stdout
Now solve the problem and return the code. | ```python
# cook your dish here
MOD=10**9+7
for _ in range(int(input())):
s=input()
ind=1
level=1
for i in range(len(s)):
if s[i]=='l':
if level%2==1:
ind=ind*2
else:
ind=ind*2-1
if s[i]=='r':
if level%2==1:
ind=ind*2+2
else:
ind=ind*2+1
level+=1
ind%=MOD
print(ind)
``` |
|
open-r1/verifiable-coding-problems-python-10k | Solve the following coding problem using the programming language python:
Mrs. Smith is trying to contact her husband, John Smith, but she forgot the secret phone number!
The only thing Mrs. Smith remembered was that any permutation of $n$ can be a secret phone number. Only those permutations that minimize secret value might be the phone of her husband.
The sequence of $n$ integers is called a permutation if it contains all integers from $1$ to $n$ exactly once.
The secret value of a phone number is defined as the sum of the length of the longest increasing subsequence (LIS) and length of the longest decreasing subsequence (LDS).
A subsequence $a_{i_1}, a_{i_2}, \ldots, a_{i_k}$ where $1\leq i_1 < i_2 < \ldots < i_k\leq n$ is called increasing if $a_{i_1} < a_{i_2} < a_{i_3} < \ldots < a_{i_k}$. If $a_{i_1} > a_{i_2} > a_{i_3} > \ldots > a_{i_k}$, a subsequence is called decreasing. An increasing/decreasing subsequence is called longest if it has maximum length among all increasing/decreasing subsequences.
For example, if there is a permutation $[6, 4, 1, 7, 2, 3, 5]$, LIS of this permutation will be $[1, 2, 3, 5]$, so the length of LIS is equal to $4$. LDS can be $[6, 4, 1]$, $[6, 4, 2]$, or $[6, 4, 3]$, so the length of LDS is $3$.
Note, the lengths of LIS and LDS can be different.
So please help Mrs. Smith to find a permutation that gives a minimum sum of lengths of LIS and LDS.
-----Input-----
The only line contains one integer $n$ ($1 \le n \le 10^5$) — the length of permutation that you need to build.
-----Output-----
Print a permutation that gives a minimum sum of lengths of LIS and LDS.
If there are multiple answers, print any.
-----Examples-----
Input
4
Output
3 4 1 2
Input
2
Output
2 1
-----Note-----
In the first sample, you can build a permutation $[3, 4, 1, 2]$. LIS is $[3, 4]$ (or $[1, 2]$), so the length of LIS is equal to $2$. LDS can be ony of $[3, 1]$, $[4, 2]$, $[3, 2]$, or $[4, 1]$. The length of LDS is also equal to $2$. The sum is equal to $4$. Note that $[3, 4, 1, 2]$ is not the only permutation that is valid.
In the second sample, you can build a permutation $[2, 1]$. LIS is $[1]$ (or $[2]$), so the length of LIS is equal to $1$. LDS is $[2, 1]$, so the length of LDS is equal to $2$. The sum is equal to $3$. Note that permutation $[1, 2]$ is also valid.
The input will be stdin and you should print your solution to stdout
Now solve the problem and return the code. | ```python
from math import sqrt
n = int(input())
k = int(sqrt(n))
b = []
last = 0
while last < n:
b.append([last + j for j in range(k)])
last = b[-1][-1] + 1
k = len(b)
for i in range(k - 1, -1, -1):
for j in b[i]:
if j < n:
print(1 + j, end=' ')
print()
``` |
|
open-r1/verifiable-coding-problems-python-10k | Solve the following coding problem using the programming language python:
Sergey recently learned about country codes - two letter strings, denoting countries. For example, BY stands for Belarus and IN stands for India. Mesmerized by this new discovery, Sergey now looks for country codes everywhere!
Sergey has recently found a string S consisting of uppercase Latin letters. He wants to find the number of different country codes that appear in S as contiguous substrings. For the purpose of this problem, consider that every 2-letter uppercase string is a valid country code.
-----Input-----
The first line of input contains an integer T denoting the number of test cases. The description of T test cases follows.
The first and only line of each test case contains a string S, consisting of uppercase Latin letters.
-----Output-----
For each test case, output a single line containing the number of different country codes appearing in the given string.
-----Constraints-----
- 1 ≤ T ≤ 100
- Subtask 1 (35 points): 2 ≤ |S| ≤ 3
- Subtask 2 (65 points): 2 ≤ |S| ≤ 104
-----Example-----
Input:2
INBY
BYBY
Output:3
2
-----Explanation-----
Example case 1. The codes are IN, NB and BY.
Example case 2. The codes are BY and YB.
The input will be stdin and you should print your solution to stdout
Now solve the problem and return the code. | ```python
#import set
t = eval(input())
while(t):
s = input()
set1 = set()
j = 0
for i in s[:-1]:
a = s[j:j+2]
set1.add(a)
j = j + 1
print(str(len(set1)) + '\n')
t= t-1
``` |
|
open-r1/verifiable-coding-problems-python-10k | Solve the following coding problem using the programming language python:
Chef loves to play with iron (Fe) and magnets (Ma). He took a row of $N$ cells (numbered $1$ through $N$) and placed some objects in some of these cells. You are given a string $S$ with length $N$ describing them; for each valid $i$, the $i$-th character of $S$ is one of the following:
- 'I' if the $i$-th cell contains a piece of iron
- 'M' if the $i$-th cell contains a magnet
- '_' if the $i$-th cell is empty
- ':' if the $i$-th cell contains a conducting sheet
- 'X' if the $i$-th cell is blocked
If there is a magnet in a cell $i$ and iron in a cell $j$, the attraction power between these cells is $P_{i,j} = K+1 - |j-i| - S_{i,j}$, where $S_{i,j}$ is the number of cells containing sheets between cells $i$ and $j$. This magnet can only attract this iron if $P_{i, j} > 0$ and there are no blocked cells between the cells $i$ and $j$.
Chef wants to choose some magnets (possibly none) and to each of these magnets, assign a piece of iron which this magnet should attract. Each piece of iron may only be attracted by at most one magnet and only if the attraction power between them is positive and there are no blocked cells between them. Find the maximum number of magnets Chef can choose.
-----Input-----
- The first line of the input contains a single integer $T$ denoting the number of test cases. The description of $T$ test cases follows.
- The first line of each test case contains two space-separated integers $N$ and $K$.
- The second line contains a single string $S$ with length $N$.
-----Output-----
For each test case, print a single line containing one integer ― the maximum number of magnets that can attract iron.
-----Constraints-----
- $1 \le T \le 2,000$
- $1 \le N \le 10^5$
- $0 \le K \le 10^5$
- $S$ contains only characters 'I', 'M', '_', ':' and 'X'
- the sum of $N$ over all test cases does not exceed $5 \cdot 10^6$
-----Subtasks-----
Subtask #1 (30 points): there are no sheets, i.e. $S$ does not contain the character ':'
Subtask #2 (70 points): original constraints
-----Example Input-----
2
4 5
I::M
9 10
MIM_XII:M
-----Example Output-----
1
2
-----Explanation-----
Example case 1: The attraction power between the only magnet and the only piece of iron is $5+1-3-2 = 1$. Note that it decreases with distance and the number of sheets.
Example case 2:
The magnets in cells $1$ and $3$ can attract the piece of iron in cell $2$, since the attraction power is $10$ in both cases. They cannot attract iron in cells $6$ or $7$ because there is a wall between them.
The magnet in cell $9$ can attract the pieces of iron in cells $7$ and $6$; the attraction power is $8$ and $7$ respectively.
The input will be stdin and you should print your solution to stdout
Now solve the problem and return the code. | ```python
# cook your dish here
# cook your dish here
for _ in range(int(input())) :
n,k=map(int,input().split())
#reading the string
s=input()
i,j=0,0
q=0
while(i<n and j<n) :
if(s[i]=='M') :
if(s[j]=='I') :
cnt=0
if(i>j) :
p=s[j:i]
cnt=p.count(':')
else :
p=s[i:j]
cnt=p.count(':')
t=k+1-abs(i-j)-cnt
if(t>0) :
q+=1
i+=1
j+=1
else:
if(i<j) :
i+=1
else:
j+=1
elif(s[j]=='X') :
j+=1
i=j
else:
j+=1
elif(s[i]=='X') :
i+=1
j=i
else:
i+=1
print(q)
``` |
|
open-r1/verifiable-coding-problems-python-10k | Solve the following coding problem using the programming language python:
You are given a permutation of natural integers from 1 to N, inclusive. Initially, the permutation is 1, 2, 3, ..., N.
You are also given M pairs of integers, where the i-th is (Li Ri). In a single turn you can choose any of these pairs (let's say with the index j) and arbitrarily shuffle the elements of our permutation on the positions from Lj to Rj, inclusive (the positions are 1-based). You are not limited in the number of turns and you can pick any pair more than once.
The goal is to obtain the permutation P, that is given to you. If it's possible, output "Possible", otherwise output "Impossible" (without quotes).
-----Input-----
The first line of the input contains an integer T denoting the number of test cases. The description of T test cases follows.
The first line of each test case contains two space separated integers N and M denoting the size of the permutation P and the number of pairs described above.
The next line contains N integers - the permutation P.
Each of the following M lines contain pair of integers Li and Ri.
-----Output-----
For each test case, output a single line containing the answer to the corresponding test case.
-----Constraints-----
- 1 ≤ T ≤ 35
- 1 ≤ N, M ≤ 100000
- 1 ≤ Li ≤ Ri ≤ N
-----Example-----
Input:
2
7 4
3 1 2 4 5 7 6
1 2
4 4
6 7
2 3
4 2
2 1 3 4
2 4
2 3
Output:
Possible
Impossible
The input will be stdin and you should print your solution to stdout
Now solve the problem and return the code. | ```python
t=int(input())
for _ in range(t):
n,m=list(map(int,input().split()))
l=list(map(int,input().split()))
k=[]
for i in range(m):
a,b=list(map(int,input().split()))
k.append([a,b])
k.sort()
c=[]
flag=1
x=k[0][0]
y=k[0][1]
for i in k[1:]:
if i[0]<=y:
y=max(y,i[1])
else:
c.append([x-1,y-1])
x=i[0]
y=i[1]
c.append([x-1,y-1])
m=[]
j=0
for i in c:
while j<i[0]:
m.append(l[j])
j+=1
x=l[i[0]:i[1]+1]
m+=sorted(x)
j=i[1]+1
while j<n:
m.append(l[j])
j+=1
if m==sorted(l):
print('Possible')
else:
print('Impossible')
``` |
|
open-r1/verifiable-coding-problems-python-10k | Solve the following coding problem using the programming language python:
You are given the set of vectors on the plane, each of them starting at the origin. Your task is to find a pair of vectors with the minimal non-oriented angle between them.
Non-oriented angle is non-negative value, minimal between clockwise and counterclockwise direction angles. Non-oriented angle is always between 0 and π. For example, opposite directions vectors have angle equals to π.
-----Input-----
First line of the input contains a single integer n (2 ≤ n ≤ 100 000) — the number of vectors.
The i-th of the following n lines contains two integers x_{i} and y_{i} (|x|, |y| ≤ 10 000, x^2 + y^2 > 0) — the coordinates of the i-th vector. Vectors are numbered from 1 to n in order of appearing in the input. It is guaranteed that no two vectors in the input share the same direction (but they still can have opposite directions).
-----Output-----
Print two integer numbers a and b (a ≠ b) — a pair of indices of vectors with the minimal non-oriented angle. You can print the numbers in any order. If there are many possible answers, print any.
-----Examples-----
Input
4
-1 0
0 -1
1 0
1 1
Output
3 4
Input
6
-1 0
0 -1
1 0
1 1
-4 -5
-4 -6
Output
6 5
The input will be stdin and you should print your solution to stdout
Now solve the problem and return the code. | ```python
from math import *
# stores counterclockwise angle between vector (1,0) and each vector in a
a = []
n = int(input())
for i in range(n):
x,y = list(map(int,input().split()))
# calculate counterclockwise angle between (1,0) and this vector
t = acos(x/sqrt(x**2+y**2))
a.append((i+1,[2*pi-t,t][y>=0],x,y))
cmp = lambda x:x[1]
a = sorted(a,key=cmp)
# construct pairs for adjacent vectors
b = []
for i in range(n):
i1,i2 = a[i][0],a[(i+1)%n][0]
x1,y1 = a[i][2:]
x2,y2 = a[(i+1)%n][2:]
inner_prod = x1*x2 + y1*y2
inner_prod *= abs(inner_prod)
norm_prod = ((x1**2+y1**2)*(x2**2+y2**2))
b.append((i1,i2,inner_prod,norm_prod))
# find the nearest vector
better = lambda p1,p2: p1[2]*p2[3]>p2[2]*p1[3]
ans = b[-1]
for i in range(n):
if better(b[i],ans):
ans = b[i]
print(ans[0],ans[1])
``` |
|
open-r1/verifiable-coding-problems-python-10k | Solve the following coding problem using the programming language python:
There is an integer sequence of length 2^N: A_0, A_1, ..., A_{2^N-1}. (Note that the sequence is 0-indexed.)
For every integer K satisfying 1 \leq K \leq 2^N-1, solve the following problem:
- Let i and j be integers. Find the maximum value of A_i + A_j where 0 \leq i < j \leq 2^N-1 and (i or j) \leq K.
Here, or denotes the bitwise OR.
-----Constraints-----
- 1 \leq N \leq 18
- 1 \leq A_i \leq 10^9
- All values in input are integers.
-----Input-----
Input is given from Standard Input in the following format:
N
A_0 A_1 ... A_{2^N-1}
-----Output-----
Print 2^N-1 lines.
In the i-th line, print the answer of the problem above for K=i.
-----Sample Input-----
2
1 2 3 1
-----Sample Output-----
3
4
5
For K=1, the only possible pair of i and j is (i,j)=(0,1), so the answer is A_0+A_1=1+2=3.
For K=2, the possible pairs of i and j are (i,j)=(0,1),(0,2).
When (i,j)=(0,2), A_i+A_j=1+3=4. This is the maximum value, so the answer is 4.
For K=3, the possible pairs of i and j are (i,j)=(0,1),(0,2),(0,3),(1,2),(1,3),(2,3) .
When (i,j)=(1,2), A_i+A_j=2+3=5. This is the maximum value, so the answer is 5.
The input will be stdin and you should print your solution to stdout
Now solve the problem and return the code. | ```python
j=n=1<<int(input())
a=[[0,int(s)]for s in input().split()]
while j>1:j>>=1;a=[sorted(a[i]+a[i^j]*(i&j>0))[-2:]for i in range(n)]
for s,f in a[1:]:j=max(j,s+f);print(j)
``` |
|
open-r1/verifiable-coding-problems-python-10k | Solve the following coding problem using the programming language python:
The MarkiT online virtual market startup wants to organize its grand opening in NIT Patna.
but they want maximum crowd for their inauguration. So the manager told this to Praveen a student in NITP who suggested them:
The first-year students come to campus every x hour,
Second-year students come to campus every y hour,
Third-year students come to campus every z hour and
Fourth-year is very busy so they don't come regularly.
So Praveen being very clever told him the no of times in n days he can have an audience of all year student (1st,2nd & 3rd) at max. So can you code what Praveen has done?
-----Input:-----
- First line will contain $T$, number of testcases. Then the testcases follow.
- Each testcase contains of a 2 line of input, first line contain one integers $N$ (No of Days).
-Next line contain 3 space separated integer the value of x y z
-----Output:-----
For each testcase, output in a single line answer the no of times audience consists of all year.
-----Constraints-----
- $1 \leq T \leq 1000$
- $1 \leq N \leq 10^8$
- $1 \leq x,y,z \leq 10^5$
-----Sample Input:-----
1
10
8 10 6
-----Sample Output:-----
2
-----EXPLANATION:-----
First favourable condition will come on 5th day and Second on 10th day.
The input will be stdin and you should print your solution to stdout
Now solve the problem and return the code. | ```python
import math
def fun(num1,num2):
if num1>num2:
a=num1
b=num2
else:
a=num2
b=num1
rem=a%b
while(rem!=0):
a=b
b=rem
rem=a%b
gcd=b
return (int((num1*num2)/gcd))
for _ in range (int(input())):
hours=int(input())*24
x,y,z=list(map(int,input().split()))
lcm=x
lcm=fun(x,y)
lcm=fun(lcm,z)
print(int(hours//lcm))
``` |
|
open-r1/verifiable-coding-problems-python-10k | Solve the following coding problem using the programming language python:
Dio Brando has set a goal for himself of becoming the richest and the most powerful being on earth.To achieve his goals he will do anything using either manipulation,seduction or plain violence.
Only one guy stands between his goal of conquering earth ,named Jotaro Kujo aka JoJo .Now Dio has a stand (visual manifestation of life energy i.e, special power ) “Za Warudo” ,which can stop time itself but needs to chant few words to activate the time stopping ability of the stand.There is a code hidden inside the words and if you can decipher the code then Dio loses his power.
In order to stop Dio ,Jotaro asks your help to decipher the code since he can’t do so while fighting Dio at the same time.
You will be given a string as input and you have to find the shortest substring which contains all the characters of itself and then make a number based on the alphabetical ordering of the characters.
For example : “bbbacdaddb”
shortest substring will be “bacd” and based on the alphabetical ordering
the answer is 2134
-----NOTE:-----
A substring is a contiguous subsegment of a string. For example, "acab" is a substring of "abacaba" (it starts in position 3 and ends in position 6), but "aa" or "d" aren't substrings of this string. So the substring of the string s from position l to position r is
S[l;r]=SlSl+1…Sr.
-----Input :-----
The first line contains $T$,number of test cases.
The second line contains a string $S$.
-----Output:-----
For each test cases ,output the number you can get from the shortest string
-----Constraints:-----
$1 \leq t \leq 100$
$1 \leq |S| \leq 10^6$
-----Test Cases:-----
-----Sample Input:-----
6
abcdfgh
urdrdrav
jojojojoj
bbbbaaa
cddfsccs
tttttttttttt
-----Sample Output:-----
1234678
2118418418122
1015
21
46193
20
-----Explanation:-----
Test case 1: abcdfgh is the shortest substring containing all char of itself and the number is 1234678
The input will be stdin and you should print your solution to stdout
Now solve the problem and return the code. | ```python
# cook your dish here
import sys
import math
from collections import Counter
from collections import OrderedDict
def inputt():
return sys.stdin.readline().strip()
def printt(n):
sys.stdout.write(str(n)+'\n')
def listt():
return [int(i) for i in inputt().split()]
def gcd(a,b):
return math.gcd(a,b)
def lcm(a,b):
return (a*b) / gcd(a,b)
from collections import defaultdict
def find_sub_string(str):
str_len = len(str)
# Count all distinct characters.
dist_count_char = len(set([x for x in str]))
ctr, start_pos, start_pos_index, min_len = 0, 0, -1, 9999999999
curr_count = defaultdict(lambda: 0)
for i in range(str_len):
curr_count[str[i]] += 1
if curr_count[str[i]] == 1:
ctr += 1
if ctr == dist_count_char:
while curr_count[str[start_pos]] > 1:
if curr_count[str[start_pos]] > 1:
curr_count[str[start_pos]] -= 1
start_pos += 1
len_window = i - start_pos + 1
if min_len > len_window:
min_len = len_window
start_pos_index = start_pos
return str[start_pos_index: start_pos_index + min_len]
t= int(inputt())
for _ in range(t):
j=[]
str1 =inputt()
s=find_sub_string(str1)
for i in s:
j.append(abs(97-ord(i))+1)
st = [str(i) for i in j]
print(''.join(st))
``` |
|
open-r1/verifiable-coding-problems-python-10k | Solve the following coding problem using the programming language python:
Recently Rocky had participated in coding competition and he is sharing one of the problem with you which he was unable to solve. Help Rocky in solving the problem.
Suppose the alphabets are arranged in a row starting with index 0$0$ from AtoZ$A to Z$.
If in a coded language A=27$A=27$ and AND=65$AND=65$.
Help Rocky to find a suitable formula for finding all the value for given test cases?
(All alphabets are in Upper case only).
-----Input:-----
First line of the input contains string s$s$.
-----Output:-----
Output the possible integer values of the given string s$s$ according to the question .
-----Constraints-----
- 1≤s≤100$1 \leq s \leq 100$
-----Sample Input:-----
A
AND
-----Sample Output:-----
27
65
The input will be stdin and you should print your solution to stdout
Now solve the problem and return the code. | ```python
# cook your dish here
s = input().strip()
start_w = 27
w_dict = {}
words = ["A", "B", "C", "D", "E", "F", "G", "H", "I", "J", "K", "L", "M", "N", "O", "P", "Q", "R", "S", "T", "U", "V", "W", "X", "Y", "Z"]
for word in words:
w_dict[word] = start_w
start_w = start_w - 1
total_wt = 0
for c in s:
total_wt = total_wt + w_dict[c]
print(total_wt)
``` |
|
open-r1/verifiable-coding-problems-python-10k | Solve the following coding problem using the programming language python:
Chef and his competitor Kefa own two restaurants located at a straight road. The position of Chef's restaurant is $X_1$, the position of Kefa's restaurant is $X_2$.
Chef and Kefa found out at the same time that a bottle with a secret recipe is located on the road between their restaurants. The position of the bottle is $X_3$.
The cooks immediately started to run to the bottle. Chef runs with speed $V_1$, Kefa with speed $V_2$.
Your task is to figure out who reaches the bottle first and gets the secret recipe (of course, it is possible that both cooks reach the bottle at the same time).
-----Input-----
- The first line of the input contains a single integer $T$ denoting the number of test cases. The description of $T$ test cases follows.
- The first and only line of each test case contains five space-separated integers $X_1$, $X_2$, $X_3$, $V_1$ and $V_2$.
-----Output-----
For each test case, print a single line containing the string "Chef" if Chef reaches the bottle first, "Kefa" if Kefa reaches the bottle first or "Draw" if Chef and Kefa reach the bottle at the same time (without quotes).
-----Constraints-----
- $1 \le T \le 10^5$
- $|X_1|, |X_2|, |X_3| \le 10^5$
- $X_1 < X_3 < X_2$
- $1 \le V_1 \le 10^5$
- $1 \le V_2 \le 10^5$
-----Example Input-----
3
1 3 2 1 2
1 5 2 1 2
1 5 3 2 2
-----Example Output-----
Kefa
Chef
Draw
-----Explanation-----
Example case 1. Chef and Kefa are on the same distance from the bottle, but Kefa has speed $2$, while Chef has speed $1$.
The input will be stdin and you should print your solution to stdout
Now solve the problem and return the code. | ```python
# cook your dish here
for t in range(int(input())):
x1,x2,x3,v1,v2=[int(x)for x in input().rstrip().split()]
t1=abs(x3-x1)/v1
t2=abs(x3-x2)/v2
if t1<t2:
print("Chef")
elif t1>t2:
print("Kefa")
elif t1==t2:
print("Draw")
else:
pass
``` |
|
open-r1/verifiable-coding-problems-python-10k | Solve the following coding problem using the programming language python:
Striver$Striver$ wants to strive hard in order to reach his goals, hence asks his mentor to give him a question for which he has to strive hard.
The mentor gives Striver$Striver$ a N$N$ X N$N$ matrix consisting of lowercase characters (′a′$'a'$ to ′z′$'z'$) and Q$Q$ queries. Every query consists of X$X$ and Y$Y$. From any position in the matrix, one can either move towards the right or towards down. He asks striver to write down all the paths from (1,1)$(1, 1)$ to (X,Y)$(X, Y)$ and find out which string has the maximum number of character ′a′$'a'$ in it and answer him the number of characters which are not 'a' in that string.
Striver wants to strive hard but also wants to impress his mentor. He asks for your help to answer Q$Q$ queries given by his mentor as fast as he can so that he can impress his mentor also. Can you help him to answer the Q queries?
-----Input:-----
- First line will contain T$T$, number of test cases. Then the test cases follow.
- First line of every test case contains a number N$N$ and Q$Q$ which denotes the dimensions of the matrix and number of queries respectively.
- N lines follow, which contains N numbers each denoting the elements of the matrix.
- Q line follow, every line contains X and Y.
-----Output:-----
For every test case, print a single integer which prints the answer to mentor's every query.
-----Constraints-----
- 1≤T≤10$1 \leq T \leq 10$
- 1≤N≤103$1 \leq N \leq 10^3$
- 1≤Q≤105$1 \leq Q \leq 10^5$
- 1≤X,Y≤N$1 \leq X, Y \leq N$
-----Sample Input:-----
1
3 2
a b a
a c d
b a b
1 3
3 3
-----Sample Output:-----
1
2
-----EXPLANATION:-----
Query-1: There is only one path from (1,1) to (1,3) i.e.,"aba" and the number of characters which are not 'a' is 1.
Query-2: The path which has the maximum number of 'a' in it is "aabab", hence non 'a' characters are 2.
The input will be stdin and you should print your solution to stdout
Now solve the problem and return the code. | ```python
a=int(input())
for _ in range(a):
c,d=list(map(int,input().split()))
crr=[[[0,0] for i in range(c+1)] for j in range(c+1)]
trr=[]
for i in range(c):
kk=list(input().split())
trr.append(kk)
for i in range(1,c+1):
for j in range(1,c+1):
if(trr[i-1][j-1]=='a'):
crr[i][j][0]=max(crr[i-1][j][0],crr[i][j-1][0])+1
if(j==1):
crr[i][j][1]=crr[i-1][j][1]+1
elif(i==1):
crr[i][j][1]=crr[i][j-1][1]+1
elif(crr[i-1][j][0]>crr[i][j-1][0]):
crr[i][j][1]=crr[i-1][j][1]+1
else:
crr[i][j][1]=crr[i][j-1][1]+1
else:
crr[i][j][0]=max(crr[i-1][j][0],crr[i][j-1][0])
if(j==1):
crr[i][j][1]=crr[i-1][j][1]+1
elif(i==1):
crr[i][j][1]=crr[i][j-1][1]+1
elif(crr[i-1][j][0]>crr[i][j-1][0]):
crr[i][j][1]=crr[i-1][j][1]+1
else:
crr[i][j][1]=crr[i][j-1][1]+1
for i in range(d):
m,n=list(map(int,input().split()))
print(crr[m][n][1]-crr[m][n][0])
``` |
|
open-r1/verifiable-coding-problems-python-10k | Solve the following coding problem using the programming language python:
The chef was chatting with his friend who was a mathematician.
Chef said "Hi !".
His friend replied that '!' is the symbol of factorial.
Chef had never heard about it and he asked more about it. Then his friend taught him how to calculate the factorial of a number.
Chef loved that But as always he got tired after calculating a few values and asked you to do it for him.
-----Input-----
N : Number of inputs
then N lines with input T
N<10
T<=200
-----Output-----
The result for the corresponding value of T
-----Example-----
Input:
3
5
4
6
Output:
120
24
720
The input will be stdin and you should print your solution to stdout
Now solve the problem and return the code. | ```python
factorials=[1]
for x in range(1,201):
factorials.append(factorials[x-1]*x)
x=int(input())
for x in range(x):
n=int(input())
print(factorials[n])
``` |
|
open-r1/verifiable-coding-problems-python-10k | Solve the following coding problem using the programming language python:
Let's call a sequence good if the sum of all its elements is $0$.
You have a sequence of integers $A_1, A_2, \ldots, A_N$. You may perform any number of operations on this sequence (including zero). In one operation, you should choose a valid index $i$ and decrease $A_i$ by $i$. Can you make the sequence good using these operations?
-----Input-----
- The first line of the input contains a single integer $T$ denoting the number of test cases. The description of $T$ test cases follows.
- The first line of each test case contains a single integer $N$.
- The second line contains $N$ space-separated integers $A_1, A_2, \ldots, A_N$.
-----Output-----
For each test case, print a single line containing the string "YES" if it is possible to make the given sequence good or "NO" if it is impossible.
-----Constraints-----
- $1 \le T \le 1,000$
- $1 \le N \le 10$
- $|A_i| \le 100$ for each valid $i$
-----Subtasks-----
Subtask #1 (10 points): $N = 1$
Subtask #2 (30 points): $N \le 2$
Subtask #3 (60 points): original constraints
-----Example Input-----
2
1
-1
2
1 2
-----Example Output-----
NO
YES
-----Explanation-----
Example case 2: We can perform two operations ― subtract $1$ from $A_1$ and $2$ from $A_2$.
The input will be stdin and you should print your solution to stdout
Now solve the problem and return the code. | ```python
n=int(input())
for i in range(n):
t=int(input())
m=list(map(int,input().split()))
p,q=0,0
if t==1:
if m[0]>=0:
print('YES')
else:
print('NO')
else:
for i in m:
if i<0:
q+=i
else:
p+=i
if p>=abs(q):
print('YES')
else:
print('NO')
``` |
|
open-r1/verifiable-coding-problems-python-10k | Solve the following coding problem using the programming language python:
The annual snake festival is upon us, and all the snakes of the kingdom have gathered to participate in the procession. Chef has been tasked with reporting on the procession, and for this he decides to first keep track of all the snakes. When he sees a snake first, it'll be its Head, and hence he will mark a 'H'. The snakes are long, and when he sees the snake finally slither away, he'll mark a 'T' to denote its tail. In the time in between, when the snake is moving past him, or the time between one snake and the next snake, he marks with '.'s.
Because the snakes come in a procession, and one by one, a valid report would be something like "..H..T...HTH....T.", or "...", or "HT", whereas "T...H..H.T", "H..T..H", "H..H..T..T" would be invalid reports (See explanations at the bottom).
Formally, a snake is represented by a 'H' followed by some (possibly zero) '.'s, and then a 'T'. A valid report is one such that it begins with a (possibly zero length) string of '.'s, and then some (possibly zero) snakes between which there can be some '.'s, and then finally ends with some (possibly zero) '.'s.
Chef had binged on the festival food and had been very drowsy. So his report might be invalid. You need to help him find out if his report is valid or not.
-----Input-----
- The first line contains a single integer, R, which denotes the number of reports to be checked. The description of each report follows after this.
- The first line of each report contains a single integer, L, the length of that report.
- The second line of each report contains a string of length L. The string contains only the characters '.', 'H', and 'T'.
-----Output-----
- For each report, output the string "Valid" or "Invalid" in a new line, depending on whether it was a valid report or not.
-----Constraints-----
- 1 ≤ R ≤ 500
- 1 ≤ length of each report ≤ 500
-----Example-----
Input:
6
18
..H..T...HTH....T.
3
...
10
H..H..T..T
2
HT
11
.T...H..H.T
7
H..T..H
Output:
Valid
Valid
Invalid
Valid
Invalid
Invalid
-----Explanation-----
"H..H..T..T" is invalid because the second snake starts before the first snake ends, which is not allowed.
".T...H..H.T" is invalid because it has a 'T' before a 'H'. A tail can come only after its head.
"H..T..H" is invalid because the last 'H' does not have a corresponding 'T'.
The input will be stdin and you should print your solution to stdout
Now solve the problem and return the code. | ```python
# cook your dish here
t=int(input())
for i in range(t):
n=int(input())
s=input().strip()
c=0
for i in range(len(s)):
if s[i]=='.':
continue
if s[i]=='H':
c+=1
if s[i]=='T':
c-=1
if c>1:
break
if c<0:
break
if c==0:
print('Valid')
else:
print('Invalid')
``` |
|
open-r1/verifiable-coding-problems-python-10k | Solve the following coding problem using the programming language python:
There is a popular apps named “Exbook” like “Facebook”. To sign up in this app , You have to make a strong password with more than 3 digits and less than 10 digits . But I am a pro hacker and so I make a Exbook hacking site . You need to login in this site to hack exbook account and then you will get a portal. You can give any user exbook login link using this site and when anyone login into exbook using your link ,you can see his/her password .
But I made a mistake and so you cannot find original password in your portal . The portal showing you by adding two in every digit . So , now you have to find out the original password of an user if I give you the password which is showing in your portal .
-----Input:-----
The first line contains a single integer t (1 ≤ t ≤ 1000) — the number of test cases.
The first line of each test case contains a single integer n which is showing in your portal . Mind it , every digit of n is greater than one .
-----Output:-----
Print , original password of user .
-----Sample Input:-----
2
3527
47269
-----Sample Output:-----
1305
25047
The input will be stdin and you should print your solution to stdout
Now solve the problem and return the code. | ```python
# cook your dish here
t=int(input())
for i in range(0,t):
p=input()
l=list(p)
for j in range(0,len(l)):
l[j]=int(l[j])
l[j]=l[j]-2
for j in range(0,len(l)):
l[j]=str(l[j])
q=''.join(l)
print(q)
``` |
|
open-r1/verifiable-coding-problems-python-10k | Solve the following coding problem using the programming language python:
Yaroslav is playing a game called "Time". The game has a timer showing the lifespan he's got left. As soon as the timer shows 0, Yaroslav's character dies and the game ends. Also, the game has n clock stations, station number i is at point (x_{i}, y_{i}) of the plane. As the player visits station number i, he increases the current time on his timer by a_{i}. The stations are for one-time use only, so if the player visits some station another time, the time on his timer won't grow.
A player spends d·dist time units to move between stations, where dist is the distance the player has covered and d is some constant. The distance between stations i and j is determined as |x_{i} - x_{j}| + |y_{i} - y_{j}|.
Initially, the player is at station number 1, and the player has strictly more than zero and strictly less than one units of time. At station number 1 one unit of money can increase the time on the timer by one time unit (you can buy only integer number of time units).
Now Yaroslav is wondering, how much money he needs to get to station n. Help Yaroslav. Consider the time to buy and to increase the timer value negligibly small.
-----Input-----
The first line contains integers n and d (3 ≤ n ≤ 100, 10^3 ≤ d ≤ 10^5) — the number of stations and the constant from the statement.
The second line contains n - 2 integers: a_2, a_3, ..., a_{n} - 1 (1 ≤ a_{i} ≤ 10^3). The next n lines contain the coordinates of the stations. The i-th of them contains two integers x_{i}, y_{i} (-100 ≤ x_{i}, y_{i} ≤ 100).
It is guaranteed that no two stations are located at the same point.
-----Output-----
In a single line print an integer — the answer to the problem.
-----Examples-----
Input
3 1000
1000
0 0
0 1
0 3
Output
2000
Input
3 1000
1000
1 0
1 1
1 2
Output
1000
The input will be stdin and you should print your solution to stdout
Now solve the problem and return the code. | ```python
n, d = map(int, input().split())
a = [0] + list(map(int, input().split())) + [0]
x = []
y = []
for i in range(n):
xx, yy = map(int, input().split())
x += [xx]
y += [yy]
b = [-1] * n
b[0] = 0
c = True
while c:
c = False
for i in range(n):
for j in range(1, n):
if i != j and b[i] != -1:
t = b[i] + (abs(x[i] - x[j]) + abs(y[i] - y[j])) * d - a[j]
if b[j] == -1 or t < b[j]:
b[j] = t
c = True
print(b[-1])
``` |
|
open-r1/verifiable-coding-problems-python-10k | Solve the following coding problem using the programming language python:
It is so boring in the summer holiday, isn't it? So Alice and Bob have invented a new game to play. The rules are as follows. First, they get a set of n distinct integers. And then they take turns to make the following moves. During each move, either Alice or Bob (the player whose turn is the current) can choose two distinct integers x and y from the set, such that the set doesn't contain their absolute difference |x - y|. Then this player adds integer |x - y| to the set (so, the size of the set increases by one).
If the current player has no valid move, he (or she) loses the game. The question is who will finally win the game if both players play optimally. Remember that Alice always moves first.
-----Input-----
The first line contains an integer n (2 ≤ n ≤ 100) — the initial number of elements in the set. The second line contains n distinct space-separated integers a_1, a_2, ..., a_{n} (1 ≤ a_{i} ≤ 10^9) — the elements of the set.
-----Output-----
Print a single line with the winner's name. If Alice wins print "Alice", otherwise print "Bob" (without quotes).
-----Examples-----
Input
2
2 3
Output
Alice
Input
2
5 3
Output
Alice
Input
3
5 6 7
Output
Bob
-----Note-----
Consider the first test sample. Alice moves first, and the only move she can do is to choose 2 and 3, then to add 1 to the set. Next Bob moves, there is no valid move anymore, so the winner is Alice.
The input will be stdin and you should print your solution to stdout
Now solve the problem and return the code. | ```python
def gcd(a, b):
while b > 0:
a, b = b, a % b
return a
n = int(input())
A = list(map(int, input().split()))
GCD = A[0]
for x in A[1:]:
GCD = gcd(GCD, x)
num = max(A) // GCD - n
if num % 2 == 0:
print("Bob")
else:
print("Alice")
``` |
|
open-r1/verifiable-coding-problems-python-10k | Solve the following coding problem using the programming language python:
Leha is a bright mathematician. Today he is investigating whether an integer is divisible by some square number or not.
He has a positive integer X represented as a product of N integers a1, a2, .... aN. He has somehow figured out that there exists some integer P such that the number X is divisible by P2, but he is not able to find such P himself. Can you find it for him? If there are more than one possible values of P possible, you can print any one of them.
-----Input-----
The first line of the input contains an integer T denoting the number of test cases. T test cases follow.
The first line of each test case contains one integer N denoting the number of intgers in presentation of X.
The second line contains N space-separated integers a1, a2, .... aN.
-----Output-----
For each test case, output a single integer P deoting the answer for this test case. Note that P must be in range from 2 to 1018 inclusive. It's guaranteed that at least one answer exists. If there are more than one possible answers, print any.
-----Constraints-----
- 1 ≤ T ≤ 5
- 1 ≤ N ≤ 100
- 1 ≤ ai ≤ 1018
-----Subtasks-----
- Subtask 1[19 points]: 1 ≤ a1*a2*...*aN ≤ 106
- Subtask 2[22 points]: 1 ≤ a1*a2*...*aN ≤ 1012
- Subtask 3[23 points]: 1 ≤ ai ≤ 1012
- Subtask 4[36 points]: no additional constraints
-----Example-----
Input:
1
3
21 11 6
Output:
3
-----Explanation-----
Example case 1. X = 21 * 11 * 6 = 1386. It's divisible by 9 which is a square number, as 9 = 32. So P = 3.
The input will be stdin and you should print your solution to stdout
Now solve the problem and return the code. | ```python
import sys
import math
r=int(input())
for v in range (0,r):
n = int(input())
x=1
arr=list(map(int,input().strip().split(" ")))
for i in range (0,n):
x=x*arr[i]
for i in range (2,100000000):
if(x%(pow(i,2))==0):
ans1=i
break
print(ans1)
``` |
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