Olympiad problems

2025 Bulgarian Winter Tournament, 11.4

Tags: Sequence , Divisibility , primes , infinite prime divisors , function , algebra , polynomial

Let $A$ be a set of $2025$ non-negative integers and $f: \mathbb{Z}_{>0} \to \mathbb{Z}_{>0}$ be a function with the following two properties: 1) For every two distinct positive integers $x,y$ there exists $a\in A$, such that $x-y$ divides $f(x+a) - f(y+a)$. 2) For every positive integer $N$ there exists a positive integer $t$ such that $f(x) \neq f(y)$ whenever $x,y \in [t, t+N]$ are distinct. Prove that there are infinitely many primes $p$ such that $p$ divides $f(x)$ for some positive integer $x$.

2010 Germany Team Selection Test, 1

Tags: function , algebra , modular arithmetic , number theory , Divisibility , IMO Shortlist

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]

1978 IMO Shortlist, 8

Tags: pigeonhole principle , number theory , Divisibility , IMO Shortlist

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?

2018 Iran MO (1st Round), 12

Tags: number theory , Divisibility

How many triples $(a,b,c)$ of positive integers strictly less than $51$ are there such that $a+b+c$ is divisible by $a, b$, and $c$?

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$

1966 IMO Shortlist, 42

Tags: Sequence , Divisibility , number theory , IMO Longlist , IMO Shortlist

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 $n.$

1999 IMO Shortlist, 6

Tags: algebra , modular arithmetic , arithmetic sequence , Divisibility , sum of digits , IMO Shortlist

Prove that for every real number $M$ there exists an infinite arithmetic progression such that: - each term is a positive integer and the common difference is not divisible by 10 - the sum of the digits of each term (in decimal representation) exceeds $M$.

2013 Brazil Team Selection Test, 3

Tags: modular arithmetic , number theory , Divisibility , IMO Shortlist

Let $x$ and $y$ be positive integers. If ${x^{2^n}}-1$ is divisible by $2^ny+1$ for every positive integer $n$, prove that $x=1$.

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.

2016 Belarus Team Selection Test, 3

Tags: IMO Shortlist , number theory , Divisibility , Hi

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.

2006 Germany Team Selection Test, 2

Tags: factorial , modular arithmetic , number theory , Divisibility , exponential , IMO Shortlist , Chinese Remainder Theorem

Find all positive integers $ n$ such that there exists a unique integer $ a$ such that $ 0\leq a < n!$ with the following property: \[ n!\mid a^n \plus{} 1 \] [i]Proposed by Carlos Caicedo, Colombia[/i]

2014 Bosnia And Herzegovina - Regional Olympiad, 4

Tags: combinatorics , Divisibility

How namy subsets with $3$ elements of set $S=\{1,2,3,...,19,20\}$ exist, such that their product is divisible by $4$.

2010 Germany Team Selection Test, 1

Tags: function , algebra , modular arithmetic , number theory , Divisibility , IMO Shortlist

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]

2000 Belarus Team Selection Test, 4.3

Tags: algebra , modular arithmetic , arithmetic sequence , Divisibility , sum of digits , IMO Shortlist

Prove that for every real number $M$ there exists an infinite arithmetic progression such that: - each term is a positive integer and the common difference is not divisible by 10 - the sum of the digits of each term (in decimal representation) exceeds $M$.

the 9th XMO, 3

Tags: number theory , Divisibility , China

A sequence $\{a_n\} $ satisfies $a_1$ is a positive integer and $a_{n+1}$ is the largest odd integer that divides $2^n-1+a_n$ for all $n\geqslant 1$. Given a positive integer $r$ which is greater than $1$. Is it possible that there exists infinitely many pairs of ordered positive integers $(m,n)$ for which $m>n$ and $a_m = ra_n$? In other words, if you successfully find [b]an[/b] $a_1$ that yields infinitely many pairs of $(m,n)$ which work fine, you win and the answer is YES. Otherwise you have to proof NO for every possible $a_1$. @below, XMO stands for Xueersi Mathematical Olympiad, where Xueersi (学而思) is a famous tutoring camp in China.

2024 Turkey EGMO TST, 2

Tags: number theory , function , Divisibility , functional equation

Find all functions $f:\mathbb{Z}^{+} \rightarrow \mathbb{Z}^{+}$ such that the conditions $\quad a) \quad a-b \mid f(a)-f(b)$ for all $a\neq b$ and $a,b \in \mathbb{Z}^{+}$ $\quad b) \quad f(\varphi(a))=\varphi(f(a))$ for all $a \in \mathbb{Z}^{+}$ where $\varphi$ is the Euler's totient function. holds

2018 China Team Selection Test, 5

Tags: number theory , binomial coefficients , Divisibility

Given a positive integer $k$, call $n$ [i]good[/i] if among $$\binom{n}{0},\binom{n}{1},\binom{n}{2},...,\binom{n}{n}$$ at least $0.99n$ of them are divisible by $k$. Show that exists some positive integer $N$ such that among $1,2,...,N$, there are at least $0.99N$ good numbers.

2012 Belarus Team Selection Test, 1

Tags: algorithm , number theory , prime numbers , Divisibility , IMO Shortlist , number of divisors

For any integer $d > 0,$ let $f(d)$ be the smallest possible integer that has exactly $d$ positive divisors (so for example we have $f(1)=1, f(5)=16,$ and $f(6)=12$). Prove that for every integer $k \geq 0$ the number $f\left(2^k\right)$ divides $f\left(2^{k+1}\right).$ [i]Proposed by Suhaimi Ramly, Malaysia[/i]

2024 Middle European Mathematical Olympiad, 4

Tags: polynomial , number theory , sum of divisors , Divisibility , factorisation

Determine all polynomials $P(x)$ with integer coefficients such that $P(n)$ is divisible by $\sigma(n)$ for all positive integers $n$. (As usual, $\sigma(n)$ denotes the sum of all positive divisors of $n$.)

2020 Iran Team Selection Test, 5

Tags: fractional part , algebra , Divisibility

For every positive integer $k>1$ prove that there exist a real number $x$ so that for every positive integer $n<1398$: $$\left\{x^n\right\}<\left\{x^{n-1}\right\} \Longleftrightarrow k\mid n.$$ [i]Proposed by Mohammad Amin Sharifi[/i]

2013 Ukraine Team Selection Test, 11

Tags: number theory , primes , Divisibility

Specified natural number $a$. Prove that there are an infinite number of prime numbers $p$ such that for some natural $n$ the number $2^{2^n} + a$ is divisible by $p$.

2010 Germany Team Selection Test, 3

Tags: modular arithmetic , number theory , IMO Shortlist , 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]

2018-IMOC, N3

Tags: number theory , Divisibility

Find all pairs of positive integers $(x,y)$ so that $$\frac{(x^2-x+1)(y^2-y+1)}{xy}\in\mathbb N.$$

2020 Moldova Team Selection Test, 6

Tags: Divisibility , algebra

Let $n$, $(n \geq3)$ be a positive integer and the polynomial $f(x)=(1+x) \cdot (1+2x) \cdot (1+3x) \cdot ... \cdot (1+nx)$ $= a_0+a_1 \cdot x+a_2 \cdot x^2+a_3 \cdot x^3+...+a_n \cdot x^n$. Show that the number $a_3$ divides the number $k=C^2_{n+1} \cdot (2 \cdot C^2_n \cdot C^2_{n+1}-3 \cdot a_2).$

2009 Serbia Team Selection Test, 1

Tags: combinatorics , permutation , Divisibility , IMO Shortlist

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]