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.

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Found problems: 545

1997 Slovenia National Olympiad, Problem 1
Tags: number theory , divisibility
Suppose that $m,n$ are integers greater than $1$ such that $m+n-1$ divides $m^2+n^2-1$. Prove that $m+n-1$ cannot be a prime number.

2004 IMO Shortlist, 6
Tags: modular arithmetic , number theory , divisibility , remainder
Given an integer ${n>1}$, denote by $P_{n}$ the product of all positive integers $x$ less than $n$ and such that $n$ divides ${x^2-1}$. For each ${n>1}$, find the remainder of $P_{n}$ on division by $n$. [i]Proposed by John Murray, Ireland[/i]

2018 Polish Junior MO First Round, 3
Tags: number theory , prime number , divisibility
Prime numbers $a, b, c$ are bigger that $3$. Show that $(a - b)(b - c)(c - a)$ is divisible by $48$.

2019 AMC 10, 9
Tags: divisibility , divisor
What is the greatest three-digit positive integer $n$ for which the sum of the first $n$ positive integers is $\underline{not}$ a divisor of the product of the first $n$ positive integers? $\textbf{(A) } 995 \qquad\textbf{(B) } 996 \qquad\textbf{(C) } 997 \qquad\textbf{(D) } 998 \qquad\textbf{(E) } 999$

2008 IMO Shortlist, 2
Tags: combinatorics , permutation , divisibility
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]

2024 Brazil Cono Sur TST, 3
Tags: quadratic residue , hensel s lemma , number theory , chinese remainder theorem , divisibility
Find all positive integers $m$ that have some multiple of the form $x^2+5y^2+2024$, with $x$ and $y$ integers.

2002 VJIMC, Problem 2
Tags: number theory , divisibility , prime
Let $p>3$ be a prime number and $n=\frac{2^{2p}-1}3$. Show that $n$ divides $2^n-2$.

2018 China Team Selection Test, 6
Tags: function , divisibility , iteration , algebra , functional equation
Let $M,a,b,r$ be non-negative integers with $a,r\ge 2$, and suppose there exists a function $f:\mathbb{Z}\rightarrow\mathbb{Z}$ satisfying the following conditions: (1) For all $n\in \mathbb{Z}$, $f^{(r)}(n)=an+b$ where $f^{(r)}$ denotes the composition of $r$ copies of $f$ (2) For all $n\ge M$, $f(n)\ge 0$ (3) For all $n>m>M$, $n-m|f(n)-f(m)$ Show that $a$ is a perfect $r$-th power.

2022 Germany Team Selection Test, 2
Tags: number theory , divisibility , cubefree
Find all positive integers $n\geq1$ such that there exists a pair $(a,b)$ of positive integers, such that $a^2+b+3$ is not divisible by the cube of any prime, and $$n=\frac{ab+3b+8}{a^2+b+3}.$$

2019 Tournament Of Towns, 3
Tags: number theory , divisibility
The product of two positive integers $m$ and $n$ is divisible by their sum. Prove that $m + n \le n^2$. (Boris Frenkin)

2010 Ukraine Team Selection Test, 6
Tags: number theory , integer , divisibility
Find all pairs of odd integers $a$ and $b$ for which there exists a natural number$ c$ such that the number $\frac{c^n+1}{2^na+b}$ is integer for all natural $n$.

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

2020 Macedonian NationŠ°l Olympiad, 1
Tags: number theory , fermat's little theorem , gcd , divisibility , prime number
Let $a, b$ be positive integers and $p, q$ be prime numbers for which $p \nmid q - 1$ and $q \mid a^p - b^p$. Prove that $q \mid a - b$.

2018 Israel Olympic Revenge, 1
Tags: prime number , divisibility , number theory , divisor
Let $n$ be a positive integer. Prove that every prime $p > 2$ that divides $(2-\sqrt{3})^n + (2+\sqrt{3})^n$ satisfy $p=1 (mod3)$

2021 Azerbaijan IMO TST, 1
Tags: number theory , sequence , divisibility
Given a positive integer $k$ show that there exists a prime $p$ such that one can choose distinct integers $a_1,a_2\cdots, a_{k+3} \in \{1, 2, \cdots ,p-1\}$ such that p divides $a_ia_{i+1}a_{i+2}a_{i+3}-i$ for all $i= 1, 2, \cdots, k$. [i]South Africa [/i]

2021 Saudi Arabia IMO TST, 10
Tags: number theory , sequence , divisibility
Given a positive integer $k$ show that there exists a prime $p$ such that one can choose distinct integers $a_1,a_2\cdots, a_{k+3} \in \{1, 2, \cdots ,p-1\}$ such that p divides $a_ia_{i+1}a_{i+2}a_{i+3}-i$ for all $i= 1, 2, \cdots, k$. [i]South Africa [/i]

2014 EGMO, 4
Tags: modular arithmetic , number theory , divisibility
Determine all positive integers $n\geq 2$ for which there exist integers $x_1,x_2,\ldots ,x_{n-1}$ satisfying the condition that if $0<i<n,0<j<n, i\neq j$ and $n$ divides $2i+j$, then $x_i<x_j$.

2015 Israel National Olympiad, 7
Tags: fibonacci , fibonacci sequence , prime number , divisibility , number theory , combinatorics
The Fibonacci sequence $F_n$ is defined by $F_0=0,F_1=1$ and the recurrence relation $F_n=F_{n-1}+F_{n-2}$ for all integers $n\geq2$. Let $p\geq3$ be a prime number. [list=a] [*] Prove that $F_{p-1}+F_{p+1}-1$ is divisible by $p$. [*] Prove that $F_{p^{k+1}-1}+F_{p^{k+1}+1}-\left(F_{p^k-1}+F_{p^k+1}\right)$ is divisible by $p^{k+1}$ for any positive integer $k$. [/list]

2017 Macedonia JBMO TST, 1
Tags: number theory , prime number , divisibility
Let $p$ be a prime number such that $3p+10$ is a sum of squares of six consecutive positive integers. Prove that $p-7$ is divisible by $36$.

2017 Bosnia Herzegovina Team Selection Test, 2
Tags: number theory , function , divisibility
Denote by $\mathbb{N}$ the set of all positive integers. Find all functions $f:\mathbb{N}\rightarrow \mathbb{N}$ such that for all positive integers $m$ and $n$, the integer $f(m)+f(n)-mn$ is nonzero and divides $mf(m)+nf(n)$. [i]Proposed by Dorlir Ahmeti, Albania[/i]

2015 Cono Sur Olympiad, 1
Tags: divisibility , number theory
Show that, for any integer $n$, the number $n^3 - 9n + 27$ is not divisible by $81$.

2012 Bogdan Stan, 2
Tags: gcd , lcm , divisibility , number theory
For any $ a\in\mathbb{Z}_{\ge 0} $ make the notation $ a\mathbb{Z}_{\ge 0} =\{ an| n\in\mathbb{Z}_{\ge 0} \} . $ Prove that the following relations are equivalent: $ \text{(1)} a\mathbb{Z}_{\ge 0} \setminus b\mathbb{Z}_{\ge 0}\subset c\mathbb{Z}_{\ge 0} \setminus d\mathbb{Z}_{\ge 0} $ $ \text{(2)} b|a\text{ or } (c|a\text{ and } \text{lcm} (a,b) |\text{lcm} (a,d)) $ [i]Marin Tolosi[/i] and [i]Cosmin Nitu[/i]

2008 Germany Team Selection Test, 2
Tags: binomial coefficient , number theory , divisibility
For every integer $ k \geq 2,$ prove that $ 2^{3k}$ divides the number \[ \binom{2^{k \plus{} 1}}{2^{k}} \minus{} \binom{2^{k}}{2^{k \minus{} 1}} \] but $ 2^{3k \plus{} 1}$ does not. [i]Author: Waldemar Pompe, Poland[/i]

2015 ELMO Problems, 1
Tags: number theory , algebra , recurrence , divisibility
Define the sequence $a_1 = 2$ and $a_n = 2^{a_{n-1}} + 2$ for all integers $n \ge 2$. Prove that $a_{n-1}$ divides $a_n$ for all integers $n \ge 2$. [i]Proposed by Sam Korsky[/i]

2022 Vietnam National Olympiad, 4
Tags: number theory , divisibility
For every pair of positive integers $(n,m)$ with $n<m$, denote $s(n,m)$ be the number of positive integers such that the number is in the range $[n,m]$ and the number is coprime with $m$. Find all positive integers $m\ge 2$ such that $m$ satisfy these condition: i) $\frac{s(n,m)}{m-n} \ge \frac{s(1,m)}{m}$ for all $n=1,2,...,m-1$; ii) $2022^m+1$ is divisible by $m^2$