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40.3k
| problem
stringlengths 10
5.15k
| ground_truth
stringlengths 1
1.22k
| solved_percentage
float64 0
100
|
|---|---|---|---|
6,000 |
Determine all integers $k\geqslant 1$ with the following property: given $k$ different colours, if each integer is coloured in one of these $k$ colours, then there must exist integers $a_1<a_2<\cdots<a_{2023}$ of the same colour such that the differences $a_2-a_1,a_3-a_2,\dots,a_{2023}-a_{2022}$ are all powers of $2$.
|
1 \text{ and } 2
| 39.84375 |
6,001 |
Consider a sequence $\{a_n\}$ of integers, satisfying $a_1=1, a_2=2$ and $a_{n+1}$ is the largest prime divisor of $a_1+a_2+\ldots+a_n$. Find $a_{100}$.
|
53
| 40.625 |
6,002 |
Find all primes $p$ such that $p^2-p+1$ is a perfect cube.
|
19
| 92.1875 |
6,003 |
There are 100 positive integers written on a board. At each step, Alex composes 50 fractions using each number written on the board exactly once, brings these fractions to their irreducible form, and then replaces the 100 numbers on the board with the new numerators and denominators to create 100 new numbers.
Find the smallest positive integer $n{}$ such that regardless of the values of the initial 100 numbers, after $n{}$ steps Alex can arrange to have on the board only pairwise coprime numbers.
|
99
| 6.25 |
6,004 |
A dance with 2018 couples takes place in Havana. For the dance, 2018 distinct points labeled $0, 1,\ldots, 2017$ are marked in a circumference and each couple is placed on a different point. For $i\geq1$, let $s_i=i\ (\textrm{mod}\ 2018)$ and $r_i=2i\ (\textrm{mod}\ 2018)$. The dance begins at minute $0$. On the $i$-th minute, the couple at point $s_i$ (if there's any) moves to point $r_i$, the couple on point $r_i$ (if there's any) drops out, and the dance continues with the remaining couples. The dance ends after $2018^2$ minutes. Determine how many couples remain at the end.
Note: If $r_i=s_i$, the couple on $s_i$ stays there and does not drop out.
|
505
| 0 |
6,005 |
If $ P(x)$ denotes a polynomial of degree $ n$ such that $ P(k)\equal{}\frac{k}{k\plus{}1}$ for $ k\equal{}0,1,2,\ldots,n$, determine $ P(n\plus{}1)$.
|
\frac{(-1)^{n+1} + (n+1)}{n+2}
| 2.34375 |
6,006 |
Let $d(n)$ denote the number of positive divisors of $n$. For positive integer $n$ we define $f(n)$ as $$f(n) = d\left(k_1\right) + d\left(k_2\right)+ \cdots + d\left(k_m\right),$$ where $1 = k_1 < k_2 < \cdots < k_m = n$ are all divisors of the number $n$. We call an integer $n > 1$ [i]almost perfect[/i] if $f(n) = n$. Find all almost perfect numbers.
|
1, 3, 18, 36
| 0 |
6,007 |
Let $n$ be a nonnegative integer. Determine the number of ways that one can choose $(n+1)^2$ sets $S_{i,j}\subseteq\{1,2,\ldots,2n\}$, for integers $i,j$ with $0\leq i,j\leq n$, such that:
[list]
[*] for all $0\leq i,j\leq n$, the set $S_{i,j}$ has $i+j$ elements; and
[*] $S_{i,j}\subseteq S_{k,l}$ whenever $0\leq i\leq k\leq n$ and $0\leq j\leq l\leq n$.
[/list]
|
(2n)! \cdot 2^{n^2}
| 0 |
6,008 |
For a given positive integer $k$ find, in terms of $k$, the minimum value of $N$ for which there is a set of $2k + 1$ distinct positive integers that has sum greater than $N$ but every subset of size $k$ has sum at most $\tfrac{N}{2}.$
|
2k^3 + 3k^2 + 3k
| 0 |
6,009 |
Mattis is hosting a badminton tournament for $40$ players on $20$ courts numbered from $1$ to $20$. The players are distributed with $2$ players on each court. In each round a winner is determined on each court. Afterwards, the player who lost on court $1$, and the player who won on court $20$ stay in place. For the remaining $38$ players, the winner on court $i$ moves to court $i + 1$ and the loser moves to court $i - 1$. The tournament continues until every player has played every other player at least once. What is the minimal number of rounds the tournament can last?
|
39
| 65.625 |
6,010 |
Find all triples $(x, y, z)$ of nonnegative integers such that
$$ x^5+x^4+1=3^y7^z $$
|
(0, 0, 0), (1, 1, 0), (2, 0, 2)
| 21.09375 |
6,011 |
A number $p$ is $perfect$ if the sum of its divisors, except $p$ is $p$. Let $f$ be a function such that:
$f(n)=0$, if n is perfect
$f(n)=0$, if the last digit of n is 4
$f(a.b)=f(a)+f(b)$
Find $f(1998)$
|
0
| 17.1875 |
6,012 |
A sequence of real numbers $a_0, a_1, . . .$ is said to be good if the following three conditions hold.
(i) The value of $a_0$ is a positive integer.
(ii) For each non-negative integer $i$ we have $a_{i+1} = 2a_i + 1 $ or $a_{i+1} =\frac{a_i}{a_i + 2} $
(iii) There exists a positive integer $k$ such that $a_k = 2014$.
Find the smallest positive integer $n$ such that there exists a good sequence $a_0, a_1, . . .$ of real numbers with the property that $a_n = 2014$.
|
60
| 0 |
6,013 |
Find a polynomial $ p\left(x\right)$ with real coefficients such that
$ \left(x\plus{}10\right)p\left(2x\right)\equal{}\left(8x\minus{}32\right)p\left(x\plus{}6\right)$
for all real $ x$ and $ p\left(1\right)\equal{}210$.
|
2(x + 4)(x - 4)(x - 8)
| 0 |
6,014 |
Find all functions $f:\mathbb{R}\to \mathbb{R}$ such that
$$f(x)+f(yf(x)+f(y))=f(x+2f(y))+xy$$for all $x,y\in \mathbb{R}$.
|
f(x) = x + 1
| 5.46875 |
6,015 |
Let $f(n)$ be the number of ways to write $n$ as a sum of powers of $2$, where we keep track of the order of the summation. For example, $f(4)=6$ because $4$ can be written as $4$, $2+2$, $2+1+1$, $1+2+1$, $1+1+2$, and $1+1+1+1$. Find the smallest $n$ greater than $2013$ for which $f(n)$ is odd.
|
2047
| 59.375 |
6,016 |
The altitudes of a triangle are $12$, $15$, and $20$. What is the area of this triangle?
|
150
| 28.125 |
6,017 |
Determine the real values of $x$ such that the triangle with sides $5$, $8$, and $x$ is obtuse.
|
(3, \sqrt{39}) \cup (\sqrt{89}, 13)
| 39.84375 |
6,018 |
In a tennis club, each member has exactly $k > 0$ friends, and a tournament is organized in rounds such that each pair of friends faces each other in matches exactly once. Rounds are played in simultaneous matches, choosing pairs until they cannot choose any more (that is, among the unchosen people, there is not a pair of friends which has its match pending). Determine the maximum number of rounds the tournament can have, depending on $k$.
|
2k - 1
| 0 |
6,019 |
Find all real numbers $a$ for which there exists a non-constant function $f :\Bbb R \to \Bbb R$ satisfying the following two equations for all $x\in \Bbb R:$
i) $f(ax) = a^2f(x)$ and
ii) $f(f(x)) = a f(x).$
|
0 \text{ and } 1
| 2.34375 |
6,020 |
Turbo the snail sits on a point on a circle with circumference $1$. Given an infinite sequence of positive real numbers $c_1, c_2, c_3, \dots$, Turbo successively crawls distances $c_1, c_2, c_3, \dots$ around the circle, each time choosing to crawl either clockwise or counterclockwise.
Determine the largest constant $C > 0$ with the following property: for every sequence of positive real numbers $c_1, c_2, c_3, \dots$ with $c_i < C$ for all $i$, Turbo can (after studying the sequence) ensure that there is some point on the circle that it will never visit or crawl across.
|
0.5
| 90.625 |
6,021 |
Let $n$ be square with 4 digits, such that all its digits are less than 6. If we add 1 to each digit the resulting number is another square. Find $n$
|
2025
| 75.78125 |
6,022 |
Find some four different natural numbers with the following property: if you add to the product of any two of them the product of the two remaining numbers. you get a prime number.
|
1, 2, 3, 5
| 35.15625 |
6,023 |
Let $Q$ be a $(2n+1) \times (2n+1)$ board. Some of its cells are colored black in such a way that every $2 \times 2$ board of $Q$ has at most $2$ black cells. Find the maximum amount of black cells that the board may have.
|
(2n+1)(n+1)
| 6.25 |
6,024 |
Triangle $ABC$ is inscribed in a circle of radius 2 with $\angle ABC \geq 90^\circ$, and $x$ is a real number satisfying the equation $x^4 + ax^3 + bx^2 + cx + 1 = 0$, where $a=BC$, $b=CA$, $c=AB$. Find all possible values of $x$.
|
$x = -\frac{1}{2} (\sqrt6 \pm \sqrt 2)$
| 0 |
6,025 |
Let $S = \left\{ 1,2,\dots,n \right\}$, where $n \ge 1$. Each of the $2^n$ subsets of $S$ is to be colored red or blue. (The subset itself is assigned a color and not its individual elements.) For any set $T \subseteq S$, we then write $f(T)$ for the number of subsets of $T$ that are blue.
Determine the number of colorings that satisfy the following condition: for any subsets $T_1$ and $T_2$ of $S$, \[ f(T_1)f(T_2) = f(T_1 \cup T_2)f(T_1 \cap T_2). \]
|
3^n + 1
| 0 |
6,026 |
Find the sum\[1+11+111+\cdots+\underbrace{111\ldots111}_{n\text{ digits}}.\]
|
\frac{10^{n+1} - 10 - 9n}{81}
| 57.8125 |
6,027 |
Find the smallest positive integer $n$ such that the $73$ fractions $\frac{19}{n+21}, \frac{20}{n+22},\frac{21}{n+23},...,\frac{91}{n+93}$ are all irreducible.
|
95
| 81.25 |
6,028 |
Find all monic polynomials $f$ with integer coefficients satisfying the following condition: there exists a positive integer $N$ such that $p$ divides $2(f(p)!)+1$ for every prime $p>N$ for which $f(p)$ is a positive integer.
|
x - 3
| 0 |
6,029 |
Let $k$ be a given positive integer. Find all triples of positive integers $a, b, c$, such that
$a + b + c = 3k + 1$,
$ab + bc + ca = 3k^2 + 2k$.
Slovakia
|
(k+1, k, k)
| 0 |
6,030 |
Consider the sequence: $x_1=19,x_2=95,x_{n+2}=\text{lcm} (x_{n+1},x_n)+x_n$, for $n>1$, where $\text{lcm} (a,b)$ means the least common multiple of $a$ and $b$. Find the greatest common divisor of $x_{1995}$ and $x_{1996}$.
|
19
| 65.625 |
6,031 |
Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that $$f(x^2y)=f(xy)+yf(f(x)+y)$$ for all real numbers $x$ and $y$.
|
f(x) = 0
| 98.4375 |
6,032 |
Find all positive integers $k<202$ for which there exist a positive integers $n$ such that
$$\bigg {\{}\frac{n}{202}\bigg {\}}+\bigg {\{}\frac{2n}{202}\bigg {\}}+\cdots +\bigg {\{}\frac{kn}{202}\bigg {\}}=\frac{k}{2}$$
|
1, 100, 101, 201
| 0 |
6,033 |
Find all functions $f: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$ such that
$$f(xf(x + y)) = yf(x) + 1$$
holds for all $x, y \in \mathbb{R}^{+}$.
|
f(x) = \frac{1}{x}
| 82.03125 |
6,034 |
Determine the largest and smallest fractions $F = \frac{y-x}{x+4y}$
if the real numbers $x$ and $y$ satisfy the equation $x^2y^2 + xy + 1 = 3y^2$.
|
$0 \leq \frac{y-x}{x+4y} \leq 4$
| 0 |
6,035 |
Find the number of pairs $(a, b)$ of positive integers with the property that the greatest common divisor of $a$ and $ b$ is equal to $1\cdot 2 \cdot 3\cdot ... \cdot50$, and the least common multiple of $a$ and $ b$ is $1^2 \cdot 2^2 \cdot 3^2\cdot ... \cdot 50^2$.
|
32768
| 43.75 |
6,036 |
As shown in the following figure, a heart is a shape consist of three semicircles with diameters $AB$, $BC$ and $AC$ such that $B$ is midpoint of the segment $AC$. A heart $\omega$ is given. Call a pair $(P, P')$ bisector if $P$ and $P'$ lie on $\omega$ and bisect its perimeter. Let $(P, P')$ and $(Q,Q')$ be bisector pairs. Tangents at points $P, P', Q$, and $Q'$ to $\omega$ construct a convex quadrilateral $XYZT$. If the quadrilateral $XYZT$ is inscribed in a circle, find the angle between lines $PP'$ and $QQ'$.
[img]https://cdn.artofproblemsolving.com/attachments/3/c/8216889594bbb504372d8cddfac73b9f56e74c.png[/img]
|
60^\circ
| 0 |
6,037 |
Find all functions $f$ from the set $\mathbb{R}$ of real numbers into $\mathbb{R}$ which satisfy for all $x, y, z \in \mathbb{R}$ the identity \[f(f(x)+f(y)+f(z))=f(f(x)-f(y))+f(2xy+f(z))+2f(xz-yz).\]
|
f(x) = 0 \text{ and } f(x) = x^2
| 80.46875 |
6,038 |
Find all functions $f: \mathbb{R} \rightarrow \mathbb{R}$ such that $$f(f(x)+y)+xf(y)=f(xy+y)+f(x)$$ for reals $x, y$.
|
f(x) = x \text{ or } f(x) = 0
| 44.53125 |
6,039 |
We consider an $n \times n$ table, with $n\ge1$. Aya wishes to color $k$ cells of this table so that that there is a unique way to place $n$ tokens on colored squares without two tokens are not in the same row or column. What is the maximum value of $k$ for which Aya's wish is achievable?
|
\frac{n(n+1)}{2}
| 0 |
6,040 |
Find all functions $f: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$, such that $f(x+f(x)+f(y))=2f(x)+y$ for all positive reals $x,y$.
|
f(x) = x
| 96.875 |
6,041 |
For each positive integer $k$ denote $C(k)$ to be sum of its distinct prime divisors. For example $C(1)=0,C(2)=2,C(45)=8$. Find all positive integers $n$ for which $C(2^n+1)=C(n)$.
|
3
| 99.21875 |
6,042 |
Determine all integral solutions of \[ a^2\plus{}b^2\plus{}c^2\equal{}a^2b^2.\]
|
(0, 0, 0)
| 60.9375 |
6,043 |
Denote $S$ as the subset of $\{1,2,3,\dots,1000\}$ with the property that none of the sums of two different elements in $S$ is in $S$. Find the maximum number of elements in $S$.
|
501
| 5.46875 |
6,044 |
Ali wants to move from point $A$ to point $B$. He cannot walk inside the black areas but he is free to move in any direction inside the white areas (not only the grid lines but the whole plane). Help Ali to find the shortest path between $A$ and $B$. Only draw the path and write its length.
[img]https://1.bp.blogspot.com/-nZrxJLfIAp8/W1RyCdnhl3I/AAAAAAAAIzQ/NM3t5EtJWMcWQS0ig0IghSo54DQUBH5hwCK4BGAYYCw/s1600/igo%2B2016.el1.png[/img]
by Morteza Saghafian
|
7 + 5\sqrt{2}
| 0 |
6,045 |
For an integer $n>2$, the tuple $(1, 2, \ldots, n)$ is written on a blackboard. On each turn, one can choose two numbers from the tuple such that their sum is a perfect square and swap them to obtain a new tuple. Find all integers $n > 2$ for which all permutations of $\{1, 2,\ldots, n\}$ can appear on the blackboard in this way.
|
n \geq 14
| 0 |
6,046 |
Find all functions $f:\mathbb{N}\rightarrow \mathbb{N}$ such that the inequality $$f(x)+yf(f(x))\le x(1+f(y))$$
holds for all positive integers $x, y$.
|
f(x) = x
| 95.3125 |
6,047 |
We colour all the sides and diagonals of a regular polygon $P$ with $43$ vertices either
red or blue in such a way that every vertex is an endpoint of $20$ red segments and $22$ blue segments.
A triangle formed by vertices of $P$ is called monochromatic if all of its sides have the same colour.
Suppose that there are $2022$ blue monochromatic triangles. How many red monochromatic triangles
are there?
|
859
| 0.78125 |
6,048 |
Let $p>3$ be a prime and let $a_1,a_2,...,a_{\frac{p-1}{2}}$ be a permutation of $1,2,...,\frac{p-1}{2}$. For which $p$ is it always possible to determine the sequence $a_1,a_2,...,a_{\frac{p-1}{2}}$ if it for all $i,j\in\{1,2,...,\frac{p-1}{2}\}$ with $i\not=j$ the residue of $a_ia_j$ modulo $p$ is known?
|
p \geq 7
| 0 |
6,049 |
Determine the maximum integer $ n $ such that for each positive integer $ k \le \frac{n}{2} $ there are two positive divisors of $ n $ with difference $ k $.
|
24
| 62.5 |
6,050 |
Find all quadruples of positive integers $(p, q, a, b)$, where $p$ and $q$ are prime numbers and $a > 1$, such that $$p^a = 1 + 5q^b.$$
|
(2, 3, 4, 1) \text{ and } (3, 2, 4, 4)
| 19.53125 |
6,051 |
Find all functions $f: (0, \infty) \to (0, \infty)$ such that
\begin{align*}
f(y(f(x))^3 + x) = x^3f(y) + f(x)
\end{align*}
for all $x, y>0$.
|
f(x) = x
| 99.21875 |
6,052 |
The numbers $1,2,\ldots,64$ are written in the squares of an $8\times 8$ chessboard, one number to each square. Then $2\times 2$ tiles are placed on the chessboard (without overlapping) so that each tile covers exactly four squares whose numbers sum to less than $100$. Find, with proof, the maximum number of tiles that can be placed on the chessboard, and give an example of a distribution of the numbers $1,2,\ldots,64$ into the squares of the chessboard that admits this maximum number of tiles.
|
12
| 5.46875 |
6,053 |
Let $\mathbb{R}^+$ be the set of positive real numbers. Find all functions $f \colon \mathbb{R}^+ \to \mathbb{R}^+$ such that, for all $x,y \in \mathbb{R}^+$,
$$f(xy+f(x))=xf(y)+2.$$
|
f(x) = x + 1
| 89.84375 |
6,054 |
Let $\mathbb{R}$ denote the set of real numbers. Find all functions $f:\mathbb{R}\rightarrow\mathbb{R}$ such that
\[f(xf(y)+y)+f(-f(x))=f(yf(x)-y)+y\]
for all $x,y\in\mathbb{R}$
|
f(x) = x + 1
| 2.34375 |
6,055 |
Find all prime numbers $ p,q$ less than 2005 and such that $ q|p^2 \plus{} 4$, $ p|q^2 \plus{} 4$.
|
(2, 2), (5, 29), (29, 5)
| 53.90625 |
6,056 |
Determine all such pairs pf positive integers $(a, b)$ such that $a + b + (gcd (a, b))^ 2 = lcm (a, b) = 2 \cdot lcm(a -1, b)$, where $lcm (a, b)$ denotes the smallest common multiple, and $gcd (a, b)$ denotes the greatest common divisor of numbers $a, b$.
|
(2, 3) \text{ and } (6, 15)
| 13.28125 |
6,057 |
Find all functions $f: \mathbb{R}^+ \to \mathbb{R}^+$ such that
$$(z + 1)f(x + y) = f(xf(z) + y) + f(yf(z) + x),$$
for all positive real numbers $x, y, z$.
|
f(x) = x
| 82.8125 |
6,058 |
Given an integer $k\geq 2$, determine all functions $f$ from the positive integers into themselves such that $f(x_1)!+f(x_2)!+\cdots f(x_k)!$ is divisibe by $x_1!+x_2!+\cdots x_k!$ for all positive integers $x_1,x_2,\cdots x_k$.
$Albania$
|
f(n) = n
| 29.6875 |
6,059 |
Find all functions $f:\mathbb{R}^+ \rightarrow \mathbb{R}^+$, such that $$f(x^{2023}+f(x)f(y))=x^{2023}+yf(x)$$ for all $x, y>0$.
|
f(x) = x
| 99.21875 |
6,060 |
Determine the maximum number of bishops that we can place in a $8 \times 8$ chessboard such that there are not two bishops in the same cell, and each bishop is threatened by at most one bishop.
Note: A bishop threatens another one, if both are placed in different cells, in the same diagonal. A board has as diagonals the $2$ main diagonals and the ones parallel to those ones.
|
20
| 0 |
6,061 |
Determine all functions $f: \mathbb{R} \to \mathbb{R}$ such that
$$ f(x^3) + f(y)^3 + f(z)^3 = 3xyz $$
for all real numbers $x$, $y$ and $z$ with $x+y+z=0$.
|
f(x) = x
| 73.4375 |
6,062 |
Find all permutations $a_1, a_2, \ldots, a_9$ of $1, 2, \ldots, 9$ such that \[ a_1+a_2+a_3+a_4=a_4+a_5+a_6+a_7= a_7+a_8+a_9+a_1 \]
and
\[ a_1^2+a_2^2+a_3^2+a_4^2=a_4^2+a_5^2+a_6^2+a_7^2= a_7^2+a_8^2+a_9^2+a_1^2 \]
|
(2, 9, 4, 5, 1, 6, 8, 3, 7)
| 0 |
6,063 |
Find all polynomials of the form $$P_n(x)=n!x^n+a_{n-1}x^{n-1}+\dots+a_1x+(-1)^n(n+1)$$ with integer coefficients, having $n$ real roots $x_1,\dots,x_n$ satisfying $k \leq x_k \leq k+1$ for $k=1, \dots,n$.
|
P_1(x) = x - 2
| 37.5 |
6,064 |
We consider positive integers $n$ having at least six positive divisors. Let the positive divisors of $n$ be arranged in a sequence $(d_i)_{1\le i\le k}$ with $$1=d_1<d_2<\dots <d_k=n\quad (k\ge 6).$$
Find all positive integers $n$ such that $$n=d_5^2+d_6^2.$$
|
500
| 85.9375 |
6,065 |
Let $n$ be a positive integer. $n$ people take part in a certain party. For any pair of the participants, either the two are acquainted with each other or they are not. What is the maximum possible number of the pairs for which the two are not acquainted but have a common acquaintance among the participants?
|
\binom{n-1}{2}
| 0.78125 |
6,066 |
Let $ABCD$ be a square with side length $1$. How many points $P$ inside the square (not on its sides) have the property that the square can be cut into $10$ triangles of equal area such that all of them have $P$ as a vertex?
|
16
| 0 |
6,067 |
Find all the positive integers less than 1000 such that the cube of the sum of its digits is equal to the square of such integer.
|
1 \text{ and } 27
| 2.34375 |
6,068 |
Equilateral triangles $ACB'$ and $BDC'$ are drawn on the diagonals of a convex quadrilateral $ABCD$ so that $B$ and $B'$ are on the same side of $AC$, and $C$ and $C'$ are on the same sides of $BD$. Find $\angle BAD + \angle CDA$ if $B'C' = AB+CD$.
|
120^\circ
| 33.59375 |
6,069 |
A $9\times 7$ rectangle is tiled with tiles of the two types: L-shaped tiles composed by three unit squares (can be rotated repeatedly with $90^\circ$) and square tiles composed by four unit squares.
Let $n\ge 0$ be the number of the $2 \times 2 $ tiles which can be used in such a tiling. Find all the values of $n$.
|
0 \text{ and } 3
| 0 |
6,070 |
Find all polynomials $P$ with integer coefficients such that $P (0)\ne 0$ and $$P^n(m)\cdot P^m(n)$$ is a square of an integer for all nonnegative integers $n, m$.
|
P(x) = x + 1
| 28.90625 |
6,071 |
For each nonnegative integer $n$ we define $A_n = 2^{3n}+3^{6n+2}+5^{6n+2}$. Find the greatest common divisor of the numbers $A_0,A_1,\ldots, A_{1999}$.
[i]Romania[/i]
|
7
| 84.375 |
6,072 |
Let $c>0$ be a given positive real and $\mathbb{R}_{>0}$ be the set of all positive reals. Find all functions $f \colon \mathbb{R}_{>0} \to \mathbb{R}_{>0}$ such that \[f((c+1)x+f(y))=f(x+2y)+2cx \quad \textrm{for all } x,y \in \mathbb{R}_{>0}.\]
|
f(x) = 2x
| 94.53125 |
6,073 |
Find all triples of primes $(p,q,r)$ satisfying $3p^{4}-5q^{4}-4r^{2}=26$.
|
(5, 3, 19)
| 45.3125 |
6,074 |
Let $\mathcal{F}$ be the set of all the functions $f : \mathcal{P}(S) \longrightarrow \mathbb{R}$ such that for all $X, Y \subseteq S$, we have $f(X \cap Y) = \min (f(X), f(Y))$, where $S$ is a finite set (and $\mathcal{P}(S)$ is the set of its subsets). Find
\[\max_{f \in \mathcal{F}}| \textrm{Im}(f) |. \]
|
n+1
| 52.34375 |
6,075 |
Find all non-negative solutions to the equation $2013^x+2014^y=2015^z$
|
(0,1,1)
| 25.78125 |
6,076 |
Determine the smallest positive integer $A$ with an odd number of digits and this property, that both $A$ and the number $B$ created by removing the middle digit of the number $A$ are divisible by $2018$.
|
100902018
| 0 |
6,077 |
A sequence $(a_n)$ of real numbers is defined by $a_0=1$, $a_1=2015$ and for all $n\geq1$, we have
$$a_{n+1}=\frac{n-1}{n+1}a_n-\frac{n-2}{n^2+n}a_{n-1}.$$
Calculate the value of $\frac{a_1}{a_2}-\frac{a_2}{a_3}+\frac{a_3}{a_4}-\frac{a_4}{a_5}+\ldots+\frac{a_{2013}}{a_{2014}}-\frac{a_{2014}}{a_{2015}}$.
|
3021
| 1.5625 |
6,078 |
Let $g:[0,1]\rightarrow \mathbb{R}$ be a continuous function and let $f_{n}:[0,1]\rightarrow \mathbb{R}$ be a
sequence of functions defined by $f_{0}(x)=g(x)$ and
$$f_{n+1}(x)=\frac{1}{x}\int_{0}^{x}f_{n}(t)dt.$$
Determine $\lim_{n\to \infty}f_{n}(x)$ for every $x\in (0,1]$.
|
g(0)
| 7.8125 |
6,079 |
Find all real $ a$, such that there exist a function $ f: \mathbb{R}\rightarrow\mathbb{R}$ satisfying the following inequality:
\[ x\plus{}af(y)\leq y\plus{}f(f(x))
\]
for all $ x,y\in\mathbb{R}$
|
a < 0 \text{ or } a = 1
| 1.5625 |
6,080 |
Find all the positive perfect cubes that are not divisible by $10$ such that the number obtained by erasing the last three digits is also a perfect cube.
|
1331 \text{ and } 1728
| 10.15625 |
6,081 |
Let $n$ be an integer greater than or equal to $1$. Find, as a function of $n$, the smallest integer $k\ge 2$ such that, among any $k$ real numbers, there are necessarily two of which the difference, in absolute value, is either strictly less than $1 / n$, either strictly greater than $n$.
|
n^2 + 2
| 18.75 |
6,082 |
Find all nonnegative integers $a, b, c$ such that
$$\sqrt{a} + \sqrt{b} + \sqrt{c} = \sqrt{2014}.$$
|
(0, 0, 2014)
| 2.34375 |
6,083 |
Find the greatest positive integer $x$ such that $23^{6+x}$ divides $2000!$
|
83
| 42.1875 |
6,084 |
A [i]permutation[/i] of the set of positive integers $[n] = \{1, 2, . . . , n\}$ is a sequence $(a_1 , a_2 , \ldots, a_n ) $ such that each element of $[n]$ appears precisely one time as a term of the sequence. For example, $(3, 5, 1, 2, 4)$ is a permutation of $[5]$. Let $P (n)$ be the number of permutations of $[n]$ for which $ka_k$ is a perfect square for all $1 \leq k \leq n$. Find with proof the smallest $n$ such that $P (n)$ is a multiple of $2010$.
|
4489
| 0 |
6,085 |
Is there a triangle with $12 \, cm^2$ area and $12$ cm perimeter?
|
\text{No}
| 50 |
6,086 |
Find the minimum positive integer $k$ such that there exists a function $f$ from the set $\Bbb{Z}$ of all integers to $\{1, 2, \ldots k\}$ with the property that $f(x) \neq f(y)$ whenever $|x-y| \in \{5, 7, 12\}$.
|
4
| 25.78125 |
6,087 |
Let $\mathbb{N}_{\geqslant 1}$ be the set of positive integers.
Find all functions $f \colon \mathbb{N}_{\geqslant 1} \to \mathbb{N}_{\geqslant 1}$ such that, for all positive integers $m$ and $n$:
\[\mathrm{GCD}\left(f(m),n\right) + \mathrm{LCM}\left(m,f(n)\right) =
\mathrm{GCD}\left(m,f(n)\right) + \mathrm{LCM}\left(f(m),n\right).\]
Note: if $a$ and $b$ are positive integers, $\mathrm{GCD}(a,b)$ is the largest positive integer that divides both $a$ and $b$, and $\mathrm{LCM}(a,b)$ is the smallest positive integer that is a multiple of both $a$ and $b$.
|
f(n) = n
| 62.5 |
6,088 |
Let $s (n)$ denote the sum of digits of a positive integer $n$. Using six different digits, we formed three 2-digits $p, q, r$ such that $$p \cdot q \cdot s(r) = p\cdot s(q) \cdot r = s (p) \cdot q \cdot r.$$ Find all such numbers $p, q, r$.
|
(12, 36, 48), (21, 63, 84)
| 0.78125 |
6,089 |
Let real $a$, $b$, and $c$ satisfy $$abc+a+b+c=ab+bc+ca+5.$$ Find the least possible value of $a^2+b^2+c^2$.
|
6
| 8.59375 |
6,090 |
Let $ABC$ be a triangle in which (${BL}$is the angle bisector of ${\angle{ABC}}$ $\left( L\in AC \right)$, ${AH}$ is an altitude of$\vartriangle ABC$ $\left( H\in BC \right)$ and ${M}$is the midpoint of the side ${AB}$. It is known that the midpoints of the segments ${BL}$ and ${MH}$ coincides. Determine the internal angles of triangle $\vartriangle ABC$.
|
60^\circ
| 0 |
6,091 |
Let $P_1,P_2,\dots,P_n$ be $n$ distinct points over a line in the plane ($n\geq2$). Consider all the circumferences with diameters $P_iP_j$ ($1\leq{i,j}\leq{n}$) and they are painted with $k$ given colors. Lets call this configuration a ($n,k$)-cloud.
For each positive integer $k$, find all the positive integers $n$ such that every possible ($n,k$)-cloud has two mutually exterior tangent circumferences of the same color.
|
n \geq 2^k + 1
| 2.34375 |
6,092 |
Alice drew a regular $2021$-gon in the plane. Bob then labeled each vertex of the $2021$-gon with a real number, in such a way that the labels of consecutive vertices differ by at most $1$. Then, for every pair of non-consecutive vertices whose labels differ by at most $1$, Alice drew a diagonal connecting them. Let $d$ be the number of diagonals Alice drew. Find the least possible value that $d$ can obtain.
|
2018
| 0 |
6,093 |
For each positive integer $n$, let $s(n)$ be the sum of the squares of the digits of $n$. For example, $s(15)=1^2+5^2=26$. Determine all integers $n\geq 1$ such that $s(n)=n$.
|
1
| 97.65625 |
6,094 |
Find all positive integers $a,b$ for which $a^4+4b^4$ is a prime number.
|
(1, 1)
| 85.9375 |
6,095 |
Find all integers $n \geq 3$ such that among any $n$ positive real numbers $a_1, a_2, \hdots, a_n$ with $\text{max}(a_1,a_2,\hdots,a_n) \leq n \cdot \text{min}(a_1,a_2,\hdots,a_n)$, there exist three that are the side lengths of an acute triangle.
|
n \geq 13
| 0 |
6,096 |
Each one of 2009 distinct points in the plane is coloured in blue or red, so that on every blue-centered unit circle there are exactly two red points. Find the gratest possible number of blue points.
|
45
| 0 |
6,097 |
Denote by $P(n)$ the greatest prime divisor of $n$. Find all integers $n\geq 2$ for which \[P(n)+\lfloor\sqrt{n}\rfloor=P(n+1)+\lfloor\sqrt{n+1}\rfloor\]
|
3
| 47.65625 |
6,098 |
Ten distinct positive real numbers are given and the sum of each pair is written (So 45 sums). Between these sums there are 5 equal numbers. If we calculate product of each pair, find the biggest number $k$ such that there may be $k$ equal numbers between them.
|
4
| 10.9375 |
6,099 |
An economist and a statistician play a game on a calculator which does only one
operation. The calculator displays only positive integers and it is used in the following
way: Denote by $n$ an integer that is shown on the calculator. A person types an integer,
$m$, chosen from the set $\{ 1, 2, . . . , 99 \}$ of the first $99$ positive integers, and if $m\%$ of the
number $n$ is again a positive integer, then the calculator displays $m\%$ of $n$. Otherwise,
the calculator shows an error message and this operation is not allowed. The game consists of doing alternatively these operations and the player that cannot do the operation
looses. How many numbers from $\{1, 2, . . . , 2019\}$ guarantee the winning strategy for the
statistician, who plays second?
For example, if the calculator displays $1200$, the economist can type $50$, giving the number
$600$ on the calculator, then the statistician can type $25$ giving the number $150$. Now, for
instance, the economist cannot type $75$ as $75\%$ of $150$ is not a positive integer, but can
choose $40$ and the game continues until one of them cannot type an allowed number
|
951
| 0 |
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