Olympiad problems

2004 IMO Shortlist, 7

Let $p$ be an odd prime and $n$ a positive integer. In the coordinate plane, eight distinct points with integer coordinates lie on a circle with diameter of length $p^{n}$. Prove that there exists a triangle with vertices at three of the given points such that the squares of its side lengths are integers divisible by $p^{n+1}$. [i]Proposed by Alexander Ivanov, Bulgaria[/i]

2015 Romania Team Selection Tests, 1

Tags: number theory , sum , divisibility , gcd

Let $a$ be an integer and $n$ a positive integer . Show that the sum : $$\sum_{k=1}^{n} a^{(k,n)}$$ is divisible by $n$ , where $(x,y)$ is the greatest common divisor of the numbers $x$ and $y$ .

2009 Belarus Team Selection Test, 2

Tags: modular arithmetic , number theory , sequence , divisibility

Let $ a_1$, $ a_2$, $ \ldots$, $ a_n$ be distinct positive integers, $ n\ge 3$. Prove that there exist distinct indices $ i$ and $ j$ such that $ a_i \plus{} a_j$ does not divide any of the numbers $ 3a_1$, $ 3a_2$, $ \ldots$, $ 3a_n$. [i]Proposed by Mohsen Jamaali, Iran[/i]

2007 IMO Shortlist, 2

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 Balkan MO, 3

Tags: number theory , divisibility , inequality , inequalities

Let $a$ and $b$ be distinct positive integers such that $3^a + 2$ is divisible by $3^b + 2$. Prove that $a > b^2$. [i]Proposed by Tynyshbek Anuarbekov, Kazakhstan[/i]

2011 All-Russian Olympiad, 1

Tags: floor function , combinatorics , divisibility

For some 2011 natural numbers, all the $\frac{2010\cdot 2011}{2}$ possible sums were written out on a board. Could it have happened that exactly one third of the written numbers were divisible by three and also exactly one third of them give a remainder of one when divided by three?

2007 IMO Shortlist, 5

Tags: function , modular arithmetic , divisibility , number theory

Find all surjective functions $ f: \mathbb{N} \to \mathbb{N}$ such that for every $ m,n \in \mathbb{N}$ and every prime $ p,$ the number $ f(m + n)$ is divisible by $ p$ if and only if $ f(m) + f(n)$ is divisible by $ p$. [i]Author: Mohsen Jamaali and Nima Ahmadi Pour Anari, Iran[/i]

1967 IMO Longlists, 17

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 $c_1c_2\ldots c_n$.

Kvant 2023, M2775

Tags: number theory , divisibility

Is there an infinite periodic sequence of digits for which the following condition condition is fulfilled: for any natural number $n{}$ a natural number divisible by $2^n{}$ can be cut from this sequence of digits (as a word)? [i]Proposed by P. Kozhevnikov[/i]

2024 VJIMC, 4

Tags: number theory , analytic number theory , divisibility

Let $p>2$ be a prime and let \[\mathcal{A}=\{n \in \mathbb{N}: 2p \mid n \text{ and } p^2\nmid n \text{ and } n \mid 3^n-1\}.\] Prove that \[\limsup_{k \to \infty} \frac{\vert \mathcal{A} \cap [1,k]\vert}{k} \le \frac{2\log 3}{p\log p}.\]

1969 IMO Shortlist, 23

Tags: divisibility , eulers function , number theory , relatively prime

$(FRA 6)$ Consider the integer $d = \frac{a^b-1}{c}$, where $a, b$, and $c$ are positive integers and $c \le a.$ Prove that the set $G$ of integers that are between $1$ and $d$ and relatively prime to $d$ (the number of such integers is denoted by $\phi(d)$) can be partitioned into $n$ subsets, each of which consists of $b$ elements. What can be said about the rational number $\frac{\phi(d)}{b}?$

2023 Romania National Olympiad, 1

Tags: divisibility , number theory

The non-zero natural number n is a perfect square. By dividing $2023$ by $n$, we obtain the remainder $223- \frac{3}{2} \cdot n$. Find the quotient of the division.

2022 IMO, 5

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

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

2021 Taiwan TST Round 1, N

Tags: number theory , sequence , divisibility

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]

2015 India Regional MathematicaI Olympiad, 3

Tags: polynomial , algebra , integer polynomial , divisibility

Let $P(x)$ be a polynomial whose coefficients are positive integers. If $P(n)$ divides $P(P(n)-2015)$ for every natural number $n$, prove that $P(-2015)=0$. [hide]One additional condition must be given that $P$ is non-constant, which even though is understood.[/hide]

2019 Ramnicean Hope, 3

Tags: algebra , polynomial , divisibility

Let be two polynoms $ P,Q\in\mathbb{C} [X] $ with degree at least $ 1, $ and such that $ P $ has only simple roots. Prove that the following affirmations are equivalent: $ \text{(i)} P\circ Q $ is divisible by $ P. $ $ \text{(ii)} $ The evaluation of $ Q $ at any root of $ P $ is a root of $ P. $ [i]Marcel Țena[/i]

2009 Germany Team Selection Test, 2

Tags: modular arithmetic , sequence , divisibility , number theory

Let $ a_1$, $ a_2$, $ \ldots$, $ a_n$ be distinct positive integers, $ n\ge 3$. Prove that there exist distinct indices $ i$ and $ j$ such that $ a_i \plus{} a_j$ does not divide any of the numbers $ 3a_1$, $ 3a_2$, $ \ldots$, $ 3a_n$. [i]Proposed by Mohsen Jamaali, Iran[/i]

2022 Greece Team Selection Test, 1

Tags: divisibility , cubefree , number theory

Find all positive integers $n\geq1$ such that there exists a pair $(a,b)$ of positive integers, such that $a^2+b+3$ is not divisible by the cube of any prime, and $$n=\frac{ab+3b+8}{a^2+b+3}.$$

2021 Turkey Team Selection Test, 1

Tags: legendre symbol , prime number , divisibility , number theory

Let $n$ be a positive integer. Prove that \[\frac{20 \cdot 5^n-2}{3^n+47}\] is not an integer.

1962 All-Soviet Union Olympiad, 12

Tags: number theory , algebra , divisibility

Given unequal integers $x, y, z$ prove that $(x-y)^5 + (y-z)^5 + (z-x)^5$ is divisible by $5(x-y)(y- z)(z-x)$.

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 $x$.)

2009 Germany Team Selection Test, 2

Tags: modular arithmetic , sequence , divisibility , number theory

Let $ a_1$, $ a_2$, $ \ldots$, $ a_n$ be distinct positive integers, $ n\ge 3$. Prove that there exist distinct indices $ i$ and $ j$ such that $ a_i \plus{} a_j$ does not divide any of the numbers $ 3a_1$, $ 3a_2$, $ \ldots$, $ 3a_n$. [i]Proposed by Mohsen Jamaali, Iran[/i]

2009 Belarus Team Selection Test, 3

Tags: combinatorics , permutation , divisibility

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]

2025 India National Olympiad, P1

Tags: recurrence , divisibility , induct

Consider the sequence defined by $a_1 = 2$, $a_2 = 3$, and \[ a_{2k+1} = 2 + 2a_k, \quad a_{2k+2} = 2 + a_k + a_{k+1}, \] for all integers $k \geq 1$. Determine all positive integers $n$ such that \[ \frac{a_n}{n} \] is an integer. Proposed by Niranjan Balachandran, SS Krishnan, and Prithwijit De.

1990 IMO Longlists, 75

Tags: number theory , divisibility , binary representation , minimization

Let $ n$ be a composite natural number and $ p$ a proper divisor of $ n.$ Find the binary representation of the smallest natural number $ N$ such that \[ \frac{(1 \plus{} 2^p \plus{} 2^{n\minus{}p})N \minus{} 1}{2^n}\] is an integer.