Olympiad problems

2003 IMO, 6

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 $q$.

2009 IMO Shortlist, 3

Tags: function , algebra , modular arithmetic , number theory , divisibility

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]

1988 IMO, 3

Tags: number theory , divisibility , diophantine equation , perfect square , vieta jumping

Let $ a$ and $ b$ be two positive integers such that $ a \cdot b \plus{} 1$ divides $ a^{2} \plus{} b^{2}$. Show that $ \frac {a^{2} \plus{} b^{2}}{a \cdot b \plus{} 1}$ is a perfect square.

2017 Korea National Olympiad, problem 5

Tags: number theory , polynomial , cubic , divisibility

Given a prime $p$, show that there exist two integers $a, b$ which satisfies the following. For all integers $m$, $m^3+ 2017am+b$ is not a multiple of $p$.

2016 India IMO Training Camp, 2

Tags: number theory , divisibility

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.

2023 Indonesia TST, 1

Tags: number theory , divisibility , prime

Find all positive integers $n>2$ such that $$ n! \mid \prod_{ p<q\le n, p,q \, \text{primes}} (p+q)$$

2003 IMO Shortlist, 7

Tags: modular arithmetic , polynomial , recurrence , sequence , divisibility , prime

The sequence $a_0$, $a_1$, $a_2,$ $\ldots$ is defined as follows: \[a_0=2, \qquad a_{k+1}=2a_k^2-1 \quad\text{for }k \geq 0.\] Prove that if an odd prime $p$ divides $a_n$, then $2^{n+3}$ divides $p^2-1$. [hide="comment"] Hi guys , Here is a nice problem: Let be given a sequence $a_n$ such that $a_0=2$ and $a_{n+1}=2a_n^2-1$ . Show that if $p$ is an odd prime such that $p|a_n$ then we have $p^2\equiv 1\pmod{2^{n+3}}$ Here are some futher question proposed by me :Prove or disprove that : 1) $gcd(n,a_n)=1$ 2) for every odd prime number $p$ we have $a_m\equiv \pm 1\pmod{p}$ where $m=\frac{p^2-1}{2^k}$ where $k=1$ or $2$ Thanks kiu si u [i]Edited by Orl.[/i] [/hide]

2016 Taiwan TST Round 2, 2

Tags: number theory , divisibility

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.

2003 IMO Shortlist, 6

Tags: number theory , polynomial , 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 $q$.

2023 Turkey Junior National Olympiad, 3

Tags: number theory , divisibility , relatively prime

Let $m,n$ be relatively prime positive integers. Prove that the numbers $$\frac{n^4+m}{m^2+n^2} \qquad \frac{n^4-m}{m^2-n^2}$$ cannot be integer at the same time.

2023 Romania EGMO TST, P2

Tags: number theory , prime number , divisibility , power of 2 , prime divisor

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.

2012 Bogdan Stan, 4

Tags: linear algebra , matrix , gcd , divisibility , induction

Prove that the elements of any natural power of a $ 2\times 2 $ special linear integer matrix are pairwise coprime, with the possible exception of the pairs that form the diagonals. [i]Vasile Pop[/i]

Russian TST 2016, P1

Tags: number theory , divisibility

Find all natural $n{}$ such that for every natural $a{}$ that is mutually prime with $n{}$, the number $a^n - 1$ is divisible by $2n^2$.

2015 India PRMO, 19

Tags: number theory , divisibility

$19.$ The digits of a positive integer $n$ are four consecutive integers in decreasing order when read from left to right. What is the sum of the possible remainders when $n$ is divided by $37 ?$

2024 Bangladesh Mathematical Olympiad, P9

Tags: number theory , factorial , divisibility , p-adic

Find all pairs of positive integers $(k, m)$ such that for any positive integer $n$, the product\[(n+m)(n+2m)\cdots(n+km)\]is divisible by $k!$.

1992 Spain Mathematical Olympiad, 1

Tags: number theory , divisibility , combinatorics , congruence

Determine the smallest number N, multiple of 83, such that N^2 has 63 positive divisors.

2025 Bulgarian Spring Mathematical Competition, 9.4

Tags: functional equation , number theory , divisibility , sophie germain identity , function , prime

Determine all functions $f: \mathbb{Z}_{>0} \to \mathbb{Z}_{>0}$ such that $f(a) + 2ab + 2f(b)$ divides $f(a)^2 + 4f(b)^2$ for any positive integers $a$ and $b$.

Russian TST 2016, P1

Tags: number theory , divisibility

Let $a{}$ and $b{}$ be natural numbers greater than one. Let $n{}$ be a natural number for which $a\mid 2^n-1$ and $b\mid 2^n+1$. Prove that there is no natural $k{}$ such that $a\mid 2^k+1$ and $b\mid 2^k-1$.

2007 IMO Shortlist, 1

Tags: divisibility , number theory

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]

2015 Bosnia Herzegovina Team Selection Test, 3

Tags: number theory , infinitely many solutions , divisibility , composite number

Prove that there exist infinitely many composite positive integers $n$ such that $n$ divides $3^{n-1}-2^{n-1}$.

1982 IMO, 1

Tags: number theory , equation , diophantine equation , divisibility

Prove that if $n$ is a positive integer such that the equation \[ x^3-3xy^2+y^3=n \] has a solution in integers $x,y$, then it has at least three such solutions. Show that the equation has no solutions in integers for $n=2891$.

2023 Abelkonkurransen Finale, 3b

Tags: number theory , divisibility

Find all integers $a$ and $b$ satisfying \begin{align*} a^6 + 1 & \mid b^{11} - 2023b^3 + 40b, \qquad \text{and}\\ a^4 - 1 & \mid b^{10} - 2023b^2 - 41. \end{align*}

2022 IMO, 5

Tags: number theory , divisibility , lte lemma , diophantine equation

Find all triples $(a,b,p)$ of positive integers with $p$ prime and \[ a^p=b!+p. \]

2016 India IMO Training Camp, 2

Tags: divisibility , number theory

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.

2004 Switzerland Team Selection Test, 8

Tags: modular arithmetic , number theory , sequence , divisibility

Let $m$ be a fixed integer greater than $1$. The sequence $x_0$, $x_1$, $x_2$, $\ldots$ is defined as follows: \[x_i = \begin{cases}2^i&\text{if }0\leq i \leq m - 1;\\\sum_{j=1}^mx_{i-j}&\text{if }i\geq m.\end{cases}\] Find the greatest $k$ for which the sequence contains $k$ consecutive terms divisible by $m$ . [i]Proposed by Marcin Kuczma, Poland[/i]