Olympiad problems

1999 IMO Shortlist, 1

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$.

1977 Germany Team Selection Test, 4

Tags: modular arithmetic , divisibility , digit , decimal representation , digit sum

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.)

2023 4th Memorial "Aleksandar Blazhevski-Cane", P6

Tags: positive integer , function , number theory , divisibility

Denote by $\mathbb{N}$ the set of positive integers. Find all functions $f:\mathbb{N} \rightarrow \mathbb{N}$ such that: [b]•[/b] For all positive integers $a> 2023^{2023}$ it holds that $f(a) \leq a$. [b]•[/b] $\frac{a^2f(b)+b^2f(a)}{f(a)+f(b)}$ is a positive integer for all $a,b \in \mathbb{N}$. [i]Proposed by Nikola Velov[/i]

2013 IFYM, Sozopol, 1

Tags: divisibility , sequence , algebra

Let $u_1=1,u_2=2,u_3=24,$ and $u_{n+1}=\frac{6u_n^2 u_{n-2}-8u_nu_{n-1}^2}{u_{n-1}u_{n-2}}, n \geq 3.$ Prove that the elements of the sequence are natural numbers and that $n\mid u_n$ for all $n$.

2022 IMO Shortlist, N2

Tags: divisibility , number theory , prime

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

2022 Korea Winter Program Practice Test, 6

Tags: number theory , divisibility

Determine all positive integers $(x_1,x_2,x_3,y_1,y_2,y_3)$ such that $y_1+ny_2^n+n^2y_3^{2n}$ divides $x_1+nx_2^n+n^2x_3^{2n}$ for all positive integer $n$.

1976 Bundeswettbewerb Mathematik, 1

Tags: number theory , divisibility

Prove that if $n$ is an odd natural number, then $1^n +2^n +\cdots +n^n$ is divisible by $n^2$.

2015 Bosnia And Herzegovina - Regional Olympiad, 1

Tags: number theory , divisibility

Find all positive integers $a$ and $b$ such that $ ab+1 \mid a^2-1$

2008 Brazil Team Selection Test, 1

Tags: modular arithmetic , number theory , divisibility

Let $b,n > 1$ be integers. Suppose that for each $k > 1$ there exists an integer $a_k$ such that $b - a^n_k$ is divisible by $k$. Prove that $b = A^n$ for some integer $A$. [i]Author: Dan Brown, Canada[/i]

2024 Kyiv City MO Round 1, Problem 3

Tags: game , number theory , combinatorics , divisibility

Let $n>1$ be a given positive integer. 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 $n$ 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 $n$ loses. Who wins if every player wants to win? Find answer for each $n>1$. [i]Proposed by Mykhailo Shtandenko, Anton Trygub[/i]

2008 Germany Team Selection Test, 3

Tags: modular arithmetic , divisibility , extremal combinatorics , additive combinatorics , number theory

Let $ X$ be a set of 10,000 integers, none of them is divisible by 47. Prove that there exists a 2007-element subset $ Y$ of $ X$ such that $ a \minus{} b \plus{} c \minus{} d \plus{} e$ is not divisible by 47 for any $ a,b,c,d,e \in Y.$ [i]Author: Gerhard Wöginger, Netherlands[/i]

2023 Polish MO Finals, 1

Tags: number theory , sequence , divisibility

Given a sequence of positive integers $a_1, a_2, a_3, \ldots$ such that for any positive integers $k$, $l$ we have $k+l ~ | ~ a_k + a_l$. Prove that for all positive integers $k > l$, $a_k - a_l$ is divisible by $k-l$.

STEMS 2021 Math Cat B, Q2

Tags: polynomial , power of 2 , prime number , divisibility , kobayashi

Determine all non-constant monic polynomials $P(x)$ with integer coefficients such that no prime $p>10^{100}$ divides any number of the form $P(2^n)$

2017 Brazil National Olympiad, 6.

Tags: number theory , divisibility , prime , group theory

[b]6.[/b] Let $a$ be a positive integer and $p$ a prime divisor of $a^3-3a+1$, with $p \neq 3$. Prove that $p$ is of the form $9k+1$ or $9k-1$, where $k$ is integer.

2025 Bangladesh Mathematical Olympiad, P8

Tags: number theory , divisibility , gcd , modular arithmetic

Let $a, b, m, n$ be positive integers such that $gcd(a, b) = 1$ and $a > 1$. Prove that if $$a^m+b^m \mid a^n+b^n$$then $m \mid n$.

2016 Silk Road, 3

Tags: floor function , coprime , divisibility , number theory

Given natural numbers $a,b$ and function $f: \mathbb{N} \to \mathbb{N} $ such that for any natural number $n, f\left( n+a \right)$ is divided by $f\left( {\left[ {\sqrt n } \right] + b} \right)$. Prove that for any natural $n$ exist $n$ pairwise distinct and pairwise relatively prime natural numbers ${{a}_{1}}$, ${{a}_{2}}$, $\ldots$, ${{a}_{n}}$ such that the number $f\left( {{a}_{i+1}} \right)$ is divided by $f\left( {{a}_{i}} \right)$ for each $i=1,2, \dots ,n-1$ . (Here $[x]$ is the integer part of number $x$, that is, the largest integer not exceeding $x$.)

2024 Czech-Polish-Slovak Junior Match, 4

Tags: divisibility , floor function , number theory

How many positive integers $n<2024$ are divisible by $\lfloor \sqrt{n}\rfloor-1$?

1982 IMO Shortlist, 16

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 Switzerland - Final Round, 6

Tags: number theory , divisibility , prime

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

1978 IMO Longlists, 23

Tags: pigeonhole principle , number theory , divisibility

Let $S$ be the set of all the odd positive integers that are not multiples of $5$ and that are less than $30m$, $m$ being an arbitrary positive integer. What is the smallest integer $k$ such that in any subset of $k$ integers from $S$ there must be two different integers, one of which divides the other?

2025 6th Memorial "Aleksandar Blazhevski-Cane", P5

Tags: number theory , divisibility , 2-adic valuation

Let $s < t$ be positive integers. Define a sequence by: $a_1 = s, a_2 = t$; $a_3$ is the smallest integer that's greater than $a_2$ and divisible by $a_1$; in general, $a_{n + 1}$ is the smallest integer greater than $a_n$ that's divisible by $a_1, a_2, ..., a_{n - 2}, a_{n - 1}$. [b]a)[/b] What is the maximum number of odd integers that can appear in such a sequence? (Justify your answer) [b]b)[/b] Prove that $a_{2025}$ is divisible by $2^{808}$, regardless of the choice of $s$ and $t$. Proposed by [i]Ilija Jovcevski[/i]

2020 Romania EGMO TST, P1

Tags: number theory , divisibility

Determine if for any positive integers $a,b,c$ there exist pairwise distinct non-negative integers $A,B,C$ which are greater than $2019$ such that $a+A,b+B$ and $c+C$ divide $ABC$.

2023-24 IOQM India, 4

Tags: number theory , divisibility , diophantine

Let $x, y$ be positive integers such that $$ x^4=(x-1)\left(y^3-23\right)-1 . $$ Find the maximum possible value of $x+y$.

1975 IMO Shortlist, 6

Tags: modular arithmetic , divisibility , digit , decimal representation , digit sum

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.)

1978 IMO Shortlist, 8

Tags: pigeonhole principle , number theory , divisibility

Let $S$ be the set of all the odd positive integers that are not multiples of $5$ and that are less than $30m$, $m$ being an arbitrary positive integer. What is the smallest integer $k$ such that in any subset of $k$ integers from $S$ there must be two different integers, one of which divides the other?