Olympiad problems

2023 Myanmar IMO Training, 1

Find all functions $f: \mathbb{N} \to \mathbb{N}$ such that $$m+f(n) \mid f(m)^2 - nf(n)$$ for all positive integers $m$ and $n$. (Here, $f(m)^2$ denotes $\left(f(m)\right)^2$.)

1990 IMO Shortlist, 23

Tags: divisibility , orders and lte , modular arithmetic , number theory

Determine all integers $ n > 1$ such that \[ \frac {2^n \plus{} 1}{n^2} \] is an integer.

2014 District Olympiad, 4

Tags: divisibility , function , functional equation , algebra

Find all functions $f:\mathbb{N}^{\ast}\rightarrow\mathbb{N}^{\ast}$ with the properties: [list=a] [*]$ f(m+n) -1 \mid f(m)+f(n),\quad \forall m,n\in\mathbb{N}^{\ast} $ [*]$ n^{2}-f(n)\text{ is a square } \;\forall n\in\mathbb{N}^{\ast} $[/list]

2015 Lusophon Mathematical Olympiad, 6

Tags: divisibility , number theory , sequence

Let $(a_n)$ be defined by: $$ a_1 = 2, \qquad a_{n+1} = a_n^3 - a_n + 1 $$ Consider positive integers $n,p$, where $p$ is an odd prime. Prove that if $p | a_n$, then $p > n$.

2024 JBMO TST - Turkey, 5

Tags: perfect square , divisibility , number theory

Find all positive integer values of $n$ such that the value of the $$\frac{2^{n!}-1}{2^n-1}$$ is a square of an integer.

2023 239 Open Mathematical Olympiad, 2

Tags: divisibility , number theory

Let $1 < a_1 < a_2 < \cdots < a_N$ be natural numbers. It is known that for any $1\leqslant i\leqslant N$ the product of all these numbers except $a_i$ increased by one, is a multiple of $a_i$. Prove that $a_1\leqslant N$.

2024 Czech-Polish-Slovak Junior Match, 4

Tags: divisibility , number theory , floor function

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

1969 IMO Shortlist, 34

Tags: divisibility , number theory

$(HUN 1)$ Let $a$ and $b$ be arbitrary integers. Prove that if $k$ is an integer not divisible by $3$, then $(a + b)^{2k}+ a^{2k} +b^{2k}$ is divisible by $a^2 +ab+ b^2$

2019 Tournament Of Towns, 3

Tags: divisibility , number theory

The product of two positive integers $m$ and $n$ is divisible by their sum. Prove that $m + n \le n^2$. (Boris Frenkin)

2007 IMO Shortlist, 4

Tags: divisibility , binomial coefficient , number theory

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]

1988 IMO Shortlist, 7

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

Let $ a$ be the greatest positive root of the equation $ x^3 \minus{} 3 \cdot x^2 \plus{} 1 \equal{} 0.$ Show that $ \left[a^{1788} \right]$ and $ \left[a^{1988} \right]$ are both divisible by 17. Here $ [x]$ denotes the integer part of $ x.$

2022 Bundeswettbewerb Mathematik, 1

Tags: divisibility , gauss sum , number theory

Five squirrels together have a supply of 2022 nuts. On the first day 2 nuts are added, on the second day 4 nuts, on the third day 6 nuts and so on, i.e. on each further day 2 nuts more are added than on the day before. At the end of any day the squirrels divide the stock among themselves. Is it possible that they all receive the same number of nuts and that no nut is left over?

2001 Slovenia National Olympiad, Problem 1

Tags: difference of squares , divisibility , algebra , special factorizations , number theory

None of the positive integers $k,m,n$ are divisible by $5$. Prove that at least one of the numbers $k^2-m^2,m^2-n^2,n^2-k^2$ is divisible by $5$.

2006 IMO Shortlist, 7

Tags: divisibility , exponential , number theory , modular arithmetic

For all positive integers $n$, show that there exists a positive integer $m$ such that $n$ divides $2^{m} + m$. [i]Proposed by Juhan Aru, Estonia[/i]

Maryland University HSMC part II, 2023.3

Tags: combination , divisibility , number theory

Let $p$ be a prime, and $n > p$ be an integer. Prove that \[ \binom{n+p-1}{p} - \binom{n}{p} \] is divisible by $n$.

2007 IMO Shortlist, 6

Tags: quadratic , vieta jumping , divisibility , number theory

Let $ k$ be a positive integer. Prove that the number $ (4 \cdot k^2 \minus{} 1)^2$ has a positive divisor of the form $ 8kn \minus{} 1$ if and only if $ k$ is even. [url=http://www.mathlinks.ro/viewtopic.php?p=894656#894656]Actual IMO 2007 Problem, posed as question 5 in the contest, which was used as a lemma in the official solutions for problem N6 as shown above.[/url] [i]Author: Kevin Buzzard and Edward Crane, United Kingdom [/i]

1999 IMO Shortlist, 1

Tags: number theory , divisibility , prime

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$.

2016 Serbia National Math Olympiad, 1

Tags: divisibility , number theory

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.

2004 IMO Shortlist, 3

Tags: divisibility , function , number theory , algebra , 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]

2024 Francophone Mathematical Olympiad, 4

Tags: divisibility , sequence , number theory

Find all integers $n \ge 2$ for which there exists $n$ integers $a_1,a_2,\dots,a_n \ge 2$ such that for all indices $i \ne j$, we have $a_i \mid a_j^2+1$.

2021 Azerbaijan IMO TST, 1

Tags: sequence , divisibility , number theory

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]

2023 Thailand TST, 1

Tags: divisibility , number theory , prime

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

2016 Indonesia TST, 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.

2024 Kyiv City MO Round 1, Problem 3

Tags: divisibility , number theory , game

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 $2023$ 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 $2023$ loses. Who wins if every player wants to win? [i]Proposed by Mykhailo Shtandenko[/i]

2015 IFYM, Sozopol, 5

Tags: divisibility , prime , number theory

Let $p>3$ be a prime number. The natural numbers $a,b,c, d$ are such that $a+b+c+d$ and $a^3+b^3+c^3+d^3$ are divisible by $p$. Prove that for all odd $n$, $a^n+b^n+c^n+d^n$ is divisible by $p$.