Found problems: 545
1969 IMO Shortlist, 23
$(FRA 6)$ Consider the integer $d = \frac{a^b-1}{c}$, where $a, b$, and $c$ are positive integers and $c \le a.$ Prove that the set $G$ of integers that are between $1$ and $d$ and relatively prime to $d$ (the number of such integers is denoted by $\phi(d)$) can be partitioned into $n$ subsets, each of which consists of $b$ elements. What can be said about the rational number $\frac{\phi(d)}{b}?$
2023 Romanian Master of Mathematics Shortlist, N1
Let $n$ be a positive integer. Let $S$ be a set of ordered pairs $(x, y)$ such that $1\leq x \leq n$ and $0 \leq y \leq n$ in each pair, and there are no pairs $(a, b)$ and $(c, d)$ of different elements in $S$ such that $a^2+b^2$ divides both
$ac+bd$ and $ad - bc$. In terms of $n$, determine the size of the largest possible set $S$.
2019 Polish MO Finals, 2
Let $p$ a prime number and $r$ an integer such that $p|r^7-1$. Prove that if there exist integers $a, b$ such that $p|r+1-a^2$ and $p|r^2+1-b^2$, then there exist an integer $c$ such that $p|r^3+1-c^2$.
2014 Bosnia And Herzegovina - Regional Olympiad, 4
How namy subsets with $3$ elements of set $S=\{1,2,3,...,19,20\}$ exist, such that their product is divisible by $4$.
2020 German National Olympiad, 4
Determine all positive integers $n$ for which there exists a positive integer $d$ with the property that $n$ is divisible by $d$ and $n^2+d^2$ is divisible by $d^2n+1$.
2016 Indonesia TST, 2
Let $m$ and $n$ be positive integers such that $m>n$. Define $x_k=\frac{m+k}{n+k}$ for $k=1,2,\ldots,n+1$. Prove that if all the numbers $x_1,x_2,\ldots,x_{n+1}$ are integers, then $x_1x_2\ldots x_{n+1}-1$ is divisible by an odd prime.
2015 Romania National Olympiad, 2
Consider a natural number $ n $ for which it exist a natural number $ k $ and $ k $ distinct primes so that $ n=p_1\cdot p_2\cdots p_k. $
[b]a)[/b] Find the number of functions $ f:\{ 1, 2,\ldots , n\}\longrightarrow\{ 1,2,\ldots ,n\} $ that have the property that $ f(1)\cdot f(2)\cdots f\left( n \right) $ divides $ n. $
[b]b)[/b] If $ n=6, $ find the number of functions $ f:\{ 1, 2,3,4,5,6\}\longrightarrow\{ 1,2,3,4,5,6\} $ that have the property that $ f(1)\cdot f(2)\cdot f(3)\cdot f(4)\cdot f(5)\cdot f(6) $ divides $ 36. $
1969 IMO Longlists, 54
$(POL 3)$ Given a polynomial $f(x)$ with integer coefficients whose value is divisible by $3$ for three integers $k, k + 1,$ and $k + 2$. Prove that $f(m)$ is divisible by $3$ for all integers $m.$
1984 IMO Shortlist, 16
Let $a,b,c,d$ be odd integers such that $0<a<b<c<d$ and $ad=bc$. Prove that if $a+d=2^k$ and $b+c=2^m$ for some integers $k$ and $m$, then $a=1$.
2010 Belarus Team Selection Test, 6.1
Let $f$ be a non-constant function from the set of positive integers into the set of positive integer, such that $a-b$ divides $f(a)-f(b)$ for all distinct positive integers $a$, $b$. Prove that there exist infinitely many primes $p$ such that $p$ divides $f(c)$ for some positive integer $c$.
[i]Proposed by Juhan Aru, Estonia[/i]
2014 Ukraine Team Selection Test, 12
Prove that for an arbitrary prime $p \ge 3$ the number of positive integers $n$, for which $p | n! +1$ does not exceed $cp^{2/3}$, where c is a constant that does not depend on $p$.
2016 Saudi Arabia BMO TST, 3
Let $ m $ and $ n $ be odd integers such that $n^2 - 1 $ is divisible by $m^2 + 1 - n^2 $.
Prove that $ |m^2 + 1 - n^2 | $ is a perfect square.
1997 IMO Shortlist, 14
Let $ b, m, n$ be positive integers such that $ b > 1$ and $ m \neq n.$ Prove that if $ b^m \minus{} 1$ and $ b^n \minus{} 1$ have the same prime divisors, then $ b \plus{} 1$ is a power of 2.
2000 Saint Petersburg Mathematical Olympiad, 10.7
We'll call a positive integer "almost prime", if it is not divisible by any prime from the interval $[3,19]$. We'll call a number "very non-prime", if it has at least 2 primes from interval $[3,19]$ dividing it. What is the greatest amount of almost prime numbers can be selected, such that the sum of any two of them is a very non-prime number?
[I]Proposed by S. Berlov, S. Ivanov[/i]
2016 Indonesia MO, 5
Given positive integers $a,b,c,d$ such that $a\mid c^d$ and $b\mid d^c$. Prove that
\[
ab\mid (cd)^{max(a,b)}
\]
1969 IMO Shortlist, 19
$(FRA 2)$ Let $n$ be an integer that is not divisible by any square greater than $1.$ Denote by $x_m$ the last digit of the number $x^m$ in the number system with base $n.$ For which integers $x$ is it possible for $x_m$ to be $0$? Prove that the sequence $x_m$ is periodic with period $t$ independent of $x.$ For which $x$ do we have $x_t = 1$. Prove that if $m$ and $x$ are relatively prime, then $0_m, 1_m, . . . , (n-1)_m$ are different numbers. Find the minimal period $t$ in terms of $n$. If n does not meet the given condition, prove that it is possible to have $x_m = 0 \neq x_1$ and that the sequence is periodic starting only from some number $k > 1.$
2015 IMO Shortlist, N3
Let $m$ and $n$ be positive integers such that $m>n$. Define $x_k=\frac{m+k}{n+k}$ for $k=1,2,\ldots,n+1$. Prove that if all the numbers $x_1,x_2,\ldots,x_{n+1}$ are integers, then $x_1x_2\ldots x_{n+1}-1$ is divisible by an odd prime.
2000 Tournament Of Towns, 4
Can one place positive integers at all vertices of a cube in such a way that for every pair of numbers connected by an edge, one will be divisible by the other , and there are no other pairs of numbers with this property?
(A Shapovalov)
2008 Germany Team Selection Test, 3
Find all surjective functions $ f: \mathbb{N} \to \mathbb{N}$ such that for every $ m,n \in \mathbb{N}$ and every prime $ p,$ the number $ f(m + n)$ is divisible by $ p$ if and only if $ f(m) + f(n)$ is divisible by $ p$.
[i]Author: Mohsen Jamaali and Nima Ahmadi Pour Anari, Iran[/i]
2015 Junior Balkan Team Selection Tests - Romania, 1
Find all the positive integers $N$ with an even number of digits with the property that if we multiply the two numbers formed by cutting the number in the middle we get a number that is a divisor of $N$ ( for example $12$ works because $1 \cdot 2$ divides $12$)
2019 Kazakhstan National Olympiad, 4
Find all positive integers $n,k,a_1,a_2,...,a_k$ so that $n^{k+1}+1$ is divisible by $(na_1+1)(na_2+1)...(na_k+1)$
1998 Iran MO (3rd Round), 1
Determine all positive integers $n$ for which there exists an integer $m$ such that ${2^{n}-1}$ is a divisor of ${m^{2}+9}$.
1999 IMO Shortlist, 7
Let $p >3$ be a prime number. For each nonempty subset $T$ of $\{0,1,2,3, \ldots , p-1\}$, let $E(T)$ be the set of all $(p-1)$-tuples $(x_1, \ldots ,x_{p-1} )$, where each $x_i \in T$ and $x_1+2x_2+ \ldots + (p-1)x_{p-1}$ is divisible by $p$ and let $|E(T)|$ denote the number of elements in $E(T)$. Prove that
\[|E(\{0,1,3\})| \geq |E(\{0,1,2\})|\]
with equality if and only if $p = 5$.
1988 IMO Longlists, 1
An integer sequence is defined by \[{ a_n = 2 a_{n-1} + a_{n-2}}, \quad (n > 1), \quad a_0 = 0, a_1 = 1.\] Prove that $2^k$ divides $a_n$ if and only if $2^k$ divides $n$.
2019 ELMO Shortlist, N4
A positive integer $b$ and a sequence $a_0,a_1,a_2,\dots$ of integers $0\le a_i<b$ is given. It is known that $a_0\neq 0$ and the sequence $\{a_i\}$ is eventually periodic but has infinitely many nonzero terms. Let $S$ be the set of positive integers $n$ so that $n\mid (a_0a_1\dots a_n)_b$. Given that $S$ is infinite, show that there are infinitely many primes that divide at least one element of $S$.
[i]Proposed by Carl Schildkraut and Holden Mui[/i]