Olympiad problems

1977 IMO, 2

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 $q^2 + r = 1977.$

2013 Brazil Team Selection Test, 3

Tags: modular arithmetic , number theory , divisibility

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 $x=1$.

2017 Bosnia and Herzegovina Team Selection Test, Problem 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]

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]

1999 Romania Team Selection Test, 10

Tags: modular arithmetic , divisibility , number theory

Determine all positive integers $n$ for which there exists an integer $m$ such that ${2^{n}-1}$ is a divisor of ${m^{2}+9}$.

1979 IMO Shortlist, 7

Tags: number theory , relatively prime , series summation , divisibility , prime

If $p$ and $q$ are natural numbers so that \[ \frac{p}{q}=1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+ \ldots -\frac{1}{1318}+\frac{1}{1319}, \] prove that $p$ is divisible with $1979$.

2010 Postal Coaching, 1

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]

2024 India National Olympiad, 3

Tags: number theory , modular arithmetic , divisibility

Let $p$ be an odd prime and $a,b,c$ be integers so that the integers $$a^{2023}+b^{2023},\quad b^{2024}+c^{2024},\quad a^{2025}+c^{2025}$$ are divisible by $p$. Prove that $p$ divides each of $a,b,c$. $\quad$ Proposed by Navilarekallu Tejaswi

2006 France Team Selection Test, 3

Tags: modular arithmetic , number theory , divisibility , chinese remainder theorem

Let $a$, $b$ be positive integers such that $b^n+n$ is a multiple of $a^n+n$ for all positive integers $n$. Prove that $a=b$. [i]Proposed by Mohsen Jamali, Iran[/i]

1993 IMO Shortlist, 3

Tags: number theory , representation , divisibility

Let $a,b,n$ be positive integers, $b > 1$ and $b^n-1\mid a.$ Show that the representation of the number $a$ in the base $b$ contains at least $n$ digits different from zero.

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 Iran Team Selection Test, 1

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.

2016 Taiwan TST Round 2, 1

Tags: factorial , number theory , divisibility

Let $a$ and $b$ be positive integers such that $a! + b!$ divides $a!b!$. Prove that $3a \ge 2b + 2$.

2022 Switzerland Team Selection Test, 1

Tags: number base , number theory , divisibility

Let $n$ be a positive integer. Prove that there exists a finite sequence $S$ consisting of only zeros and ones, satisfying the following property: for any positive integer $d \geq 2$, when $S$ is interpreted in base $d$, the resulting number is non-zero and divisible by $n$. [i]Remark: The sequence $S=s_ks_{k-1} \cdots s_1s_0$ interpreted in base $d$ is the number $\sum_{i=0}^{k}s_id^i$[/i]

2021 Polish Junior MO Finals, 5

Tags: digit , divisibility , number theory

Natural numbers $a$, $b$ are written in decimal using the same digits (i.e. every digit from 0 to 9 appears the same number of times in $a$ and in $b$). Prove that if $a+b=10^{1000}$ then both numbers $a$ and $b$ are divisible by $10$.

2016 Serbia National Math Olympiad, 1

Tags: number theory , divisibility

Let $n>1$ be an integer. Prove that there exist $m>n^n $ such that $\frac {n^m-m^n}{m+n} $ is a positive integer.

2005 India IMO Training Camp, 2

Tags: function , number theory , algebra , divisibility , functional equation

Find all functions $ f: \mathbb{N^{*}}\to \mathbb{N^{*}}$ satisfying \[ \left(f^{2}\left(m\right)+f\left(n\right)\right) \mid \left(m^{2}+n\right)^{2}\] for any two positive integers $ m$ and $ n$. [i]Remark.[/i] The abbreviation $ \mathbb{N^{*}}$ stands for the set of all positive integers: $ \mathbb{N^{*}}=\left\{1,2,3,...\right\}$. By $ f^{2}\left(m\right)$, we mean $ \left(f\left(m\right)\right)^{2}$ (and not $ f\left(f\left(m\right)\right)$). [i]Proposed by Mohsen Jamali, Iran[/i]

2019 Switzerland - Final Round, 8

Tags: number theory , divisibility , sum of divisors

An integer $n\ge2$ is called [i]resistant[/i], if it is coprime to the sum of all its divisors (including $1$ and $n$). Determine the maximum number of consecutive resistant numbers. For instance: * $n=5$ has sum of divisors $S=6$ and hence is resistant. * $n=6$ has sum of divisors $S=12$ and hence is not resistant. * $n=8$ has sum of divisors $S=15$ and hence is resistant. * $n=18$ has sum of divisors $S=39$ and hence is not resistant.

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]

2010 Germany Team Selection Test, 3

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]

2019 Poland - Second Round, 4

Tags: number theory , divisibility , positive integer

Let $a_1, a_2, \ldots, a_n$ ($n\ge 3$) be positive integers such that $gcd(a_1, a_2, \ldots, a_n)=1$ and for each $i\in \lbrace 1,2,\ldots, n \rbrace$ we have $a_i|a_1+a_2+\ldots+a_n$. Prove that $a_1a_2\ldots a_n | (a_1+a_2+\ldots+a_n)^{n-2}$.

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 $c_1c_2\ldots c_n$.

1967 IMO Longlists, 17

Tags: divisibility , binomial coefficient , number theory

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 $c_1c_2\ldots c_n$.

1993 IMO Shortlist, 2

Tags: modular arithmetic , number theory , divisibility , prime number , composite number

A natural number $n$ is said to have the property $P,$ if, for all $a, n^2$ divides $a^n - 1$ whenever $n$ divides $a^n - 1.$ a.) Show that every prime number $n$ has property $P.$ b.) Show that there are infinitely many composite numbers $n$ that possess property $P.$

2008 Brazil Team Selection Test, 1

Tags: modular arithmetic , divisibility , number theory

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]