This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

AND:
OR:
NO:

Found problems: 545

2022 3rd Memorial "Aleksandar Blazhevski-Cane", P4
Tags: set , disjoint subsets , divisibility , number theory
Find all positive integers $n$ such that the set $S=\{1,2,3, \dots 2n\}$ can be divided into $2$ disjoint subsets $S_1$ and $S_2$, i.e. $S_1 \cap S_2 = \emptyset$ and $S_1 \cup S_2 = S$, such that each one of them has $n$ elements, and the sum of the elements of $S_1$ is divisible by the sum of the elements in $S_2$. [i]Proposed by Viktor Simjanoski[/i]

2024 Bangladesh Mathematical Olympiad, P1
Tags: number theory , diophantine equation , divisibility , factorization
Find all non-negative integers $x, y$ such that\[x^3y+x+y=xy+2xy^2\]

2008 Brazil Team Selection Test, 1
Tags: number theory , divisibility
Find all pairs of natural numbers $ (a, b)$ such that $ 7^a \minus{} 3^b$ divides $ a^4 \plus{} b^2$. [i]Author: Stephan Wagner, Austria[/i]

2025 VJIMC, 1
Tags: prime , divisibility , number theory
Let $a\geq 2$ be an integer. Prove that there exists a positive integer $b$ with the following property: For each positive integer $n$, there is a prime number $p$ (possibly depending on $a,b,n$) such that $a^n + b$ is divisible by $p$, but not divisible by $p^2$.

2009 Brazil Team Selection Test, 3
Tags: number theory , sequence , divisibility , modular arithmetic
Let $ a_1$, $ a_2$, $ \ldots$, $ a_n$ be distinct positive integers, $ n\ge 3$. Prove that there exist distinct indices $ i$ and $ j$ such that $ a_i \plus{} a_j$ does not divide any of the numbers $ 3a_1$, $ 3a_2$, $ \ldots$, $ 3a_n$. [i]Proposed by Mohsen Jamaali, Iran[/i]

2023 JBMO TST - Turkey, 4
Tags: divisibility , number theory
For a prime number $p$. Can the number of n positive integers that make the expression \[\dfrac{n^3+np+1}{n+p+1}\] an integer be $777$?

2018 Dutch BxMO TST, 3
Tags: number theory , prime number , divisibility
Let $p$ be a prime number. Prove that it is possible to choose a permutation $a_1, a_2,...,a_p$ of $1,2,...,p$ such that the numbers $a_1, a_1a_2, a_1a_2a_3,..., a_1a_2a_3...a_p$ all have different remainder upon division by $p$.

2000 IMO Shortlist, 4
Tags: algebra , divisibility , polynomial , number theory
Find all triplets of positive integers $ (a,m,n)$ such that $ a^m \plus{} 1 \mid (a \plus{} 1)^n$.

2022 IMO Shortlist, N2
Tags: prime , number theory , divisibility
Find all positive integers $n>2$ such that $$ n! \mid \prod_{ p<q\le n, p,q \, \text{primes}} (p+q)$$

2024 Kyiv City MO Round 1, Problem 3
Tags: number theory , game , divisibility
Petro and Vasyl play the following game. They take turns making moves and Petro goes first. In one turn, a player chooses one of the numbers from $1$ to $2024$ that wasn't selected before and writes it on the board. The first player after whose turn the product of the numbers on the board will be divisible by $2024$ loses. Who wins if every player wants to win? [i]Proposed by Mykhailo Shtandenko[/i]

1977 IMO, 2
Tags: additive number theory , remainder , divisibility , number theory
Let $a,b$ be two natural numbers. When we divide $a^2+b^2$ by $a+b$, we the the remainder $r$ and the quotient $q.$ Determine all pairs $(a, b)$ for which $q^2 + r = 1977.$

1999 Mongolian Mathematical Olympiad, Problem 1
Tags: divisibility , number theory
Prove that for any positive integer $k$ there exist infinitely many positive integers $m$ such that $3^k\mid m^3+10$.

1999 IMO, 4
Tags: number theory , divisibility , prime
Find all the pairs of positive integers $(x,p)$ such that p is a prime, $x \leq 2p$ and $x^{p-1}$ is a divisor of $ (p-1)^{x}+1$.

1989 Bundeswettbewerb Mathematik, 1
Tags: number theory , algebra , divisibility , degree , polynomial
Determine the polynomial $$f(x) = x^k + a_{k-1} x^{k-1}+\cdots +a_1 x +a_0 $$ of smallest degree such that $a_i \in \{-1,0,1\}$ for $0\leq i \leq k-1$ and $f(n)$ is divisible by $30$ for all positive integers $n$.

2014 IMO Shortlist, N6
Tags: divisibility , modulo arithmetic , counting in two ways , integer polynomial , number theory
Let $a_1 < a_2 < \cdots <a_n$ be pairwise coprime positive integers with $a_1$ being prime and $a_1 \ge n + 2$. On the segment $I = [0, a_1 a_2 \cdots a_n ]$ of the real line, mark all integers that are divisible by at least one of the numbers $a_1 , \ldots , a_n$ . These points split $I$ into a number of smaller segments. Prove that the sum of the squares of the lengths of these segments is divisible by $a_1$. [i]Proposed by Serbia[/i]

2015 Indonesia MO, 5
Tags: divisibility , number theory
Given positive integers $a,b,c,d$ such that $a\mid c^d$ and $b\mid d^c$. Prove that \[ ab\mid (cd)^{max(a,b)} \]

1966 IMO Shortlist, 42
Tags: number theory , sequence , divisibility
Given a finite sequence of integers $a_{1},$ $a_{2},$ $...,$ $a_{n}$ for $n\geq 2.$ Show that there exists a subsequence $a_{k_{1}},$ $a_{k_{2}},$ $...,$ $a_{k_{m}},$ where $1\leq k_{1}\leq k_{2}\leq...\leq k_{m}\leq n,$ such that the number $a_{k_{1}}^{2}+a_{k_{2}}^{2}+...+a_{k_{m}}^{2}$ is divisible by $n.$ [b]Note by Darij:[/b] Of course, the $1\leq k_{1}\leq k_{2}\leq ...\leq k_{m}\leq n$ should be understood as $1\leq k_{1}<k_{2}<...<k_{m}\leq n;$ else, we could take $m=n$ and $k_{1}=k_{2}=...=k_{m},$ so that the number $a_{k_{1}}^{2}+a_{k_{2}}^{2}+...+a_{k_{m}}^{2}=n^{2}a_{k_{1}}^{2}$ will surely be divisible by $n.$

2009 Serbia National Math Olympiad, 4
Tags: permutation , divisibility , combinatorics
Let $n \in \mathbb N$ and $A_n$ set of all permutations $(a_1, \ldots, a_n)$ of the set $\{1, 2, \ldots , n\}$ for which \[k|2(a_1 + \cdots+ a_k), \text{ for all } 1 \leq k \leq n.\] Find the number of elements of the set $A_n$. [i]Proposed by Vidan Govedarica, Serbia[/i]

1977 Germany Team Selection Test, 4
Tags: divisibility , digit , decimal representation , digit sum , modular arithmetic
When $4444^{4444}$ is written in decimal notation, the sum of its digits is $ A.$ Let $B$ be the sum of the digits of $A.$ Find the sum of the digits of $ B.$ ($A$ and $B$ are written in decimal notation.)

2003 IMO Shortlist, 6
Tags: polynomial , number theory , divisibility
Let $p$ be a prime number. Prove that there exists a prime number $q$ such that for every integer $n$, the number $n^p-p$ is not divisible by $q$.

2010 Peru IMO TST, 9
Tags: modular arithmetic , number theory , sequence , divisibility
Find all positive integers $n$ such that there exists a sequence of positive integers $a_1$, $a_2$,$\ldots$, $a_n$ satisfying: \[a_{k+1}=\frac{a_k^2+1}{a_{k-1}+1}-1\] for every $k$ with $2\leq k\leq n-1$. [i]Proposed by North Korea[/i]

2013 North Korea Team Selection Test, 3
Tags: modular arithmetic , number theory , quadratic , north korea , divisibility
Find all $ a, b, c \in \mathbb{Z} $, $ c \ge 0 $ such that $ a^n + 2^n | b^n + c $ for all positive integers $ n $ where $ 2ab $ is non-square.

2015 IMO Shortlist, N2
Tags: number theory , factorial , divisibility
Let $a$ and $b$ be positive integers such that $a! + b!$ divides $a!b!$. Prove that $3a \ge 2b + 2$.

1967 IMO, 3
Tags: number theory , divisibility , binomial coefficient
Let $k,m,n$ be natural numbers such that $m+k+1$ is a prime greater than $n+1$. Let $c_s=s(s+1)$. Prove that \[(c_{m+1}-c_k)(c_{m+2}-c_k)\ldots(c_{m+n}-c_k)\] is divisible by the product $c_1c_2\ldots c_n$.

2016 Irish Math Olympiad, 1
Tags: divisibility , number theory
If the three-digit number $ABC$ is divisible by $27$, prove that the three-digit numbers $BCA$ and $CAB$ are also divisible by $27$.