Olympiad problems

1969 IMO Longlists, 34

$(HUN 1)$ 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 $a^2 +ab+ b^2$

2005 IMO Shortlist, 4

Tags: factorial , modular arithmetic , number theory , divisibility , exponential , 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]

2012 Bosnia And Herzegovina - Regional Olympiad, 2

Tags: number theory , divisibility

Let $a$, $b$, $c$ and $d$ be integers such that $ac$, $bd$ and $bc+ad$ are divisible with positive integer $m$. Show that numbers $bc$ and $ad$ are divisible with $m$

1999 Mongolian Mathematical Olympiad, Problem 1

Tags: number theory , divisibility

Prove that for any positive integer $k$ there exist infinitely many positive integers $m$ such that $3^k\mid m^3+10$.

2020 USA TSTST, 4

Tags: number theory , divisibility , floor function

Find all pairs of positive integers $(a,b)$ satisfying the following conditions: [list] [*] $a$ divides $b^4+1$, [*] $b$ divides $a^4+1$, [*] $\lfloor\sqrt{a}\rfloor=\lfloor \sqrt{b}\rfloor$. [/list] [i]Yang Liu[/i]

2005 China Team Selection Test, 1

Tags: number theory , prime number , divisibility , power of 2 , prime divisor

Let $ b, m, n$ be positive integers such that $ b > 1$ and $ m \neq n.$ Prove that if $ b^m \minus{} 1$ and $ b^n \minus{} 1$ have the same prime divisors, then $ b \plus{} 1$ is a power of 2.

2014 IMO Shortlist, N6

Tags: number theory , divisibility , modulo arithmetic , counting in two ways , integer polynomial

Let $a_1 < a_2 < \cdots <a_n$ be pairwise coprime positive integers with $a_1$ being prime and $a_1 \ge n + 2$. On the segment $I = [0, a_1 a_2 \cdots a_n ]$ of the real line, mark all integers that are divisible by at least one of the numbers $a_1 , \ldots , a_n$ . These points split $I$ into a number of smaller segments. Prove that the sum of the squares of the lengths of these segments is divisible by $a_1$. [i]Proposed by Serbia[/i]

2003 IMO Shortlist, 1

Tags: modular arithmetic , number theory , sequence , divisibility

Let $m$ be a fixed integer greater than $1$. The sequence $x_0$, $x_1$, $x_2$, $\ldots$ is defined as follows: \[x_i = \begin{cases}2^i&\text{if }0\leq i \leq m - 1;\\\sum_{j=1}^mx_{i-j}&\text{if }i\geq m.\end{cases}\] Find the greatest $k$ for which the sequence contains $k$ consecutive terms divisible by $m$ . [i]Proposed by Marcin Kuczma, Poland[/i]

1969 IMO Shortlist, 23

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

$(FRA 6)$ 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 $\frac{\phi(d)}{b}?$

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]

2023 239 Open Mathematical Olympiad, 2

Tags: number theory , divisibility

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 $a_1\leqslant N$.

2018 India National Olympiad, 4

Tags: algebra , divisibility , roots of a polynomial , polynomial

Find all polynomials with real coefficients $P(x)$ such that $P(x^2+x+1)$ divides $P(x^3-1)$.

PEN A Problems, 103

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.)

2016 Greece Team Selection Test, 1

Tags: number theory , sequence , divisibility

Given is the sequence $(a_n)_{n\geq 0}$ which is defined as follows:$a_0=3$ and $a_{n+1}-a_n=n(a_n-1) \ , \ \forall n\geq 0$. Determine all positive integers $m$ such that $\gcd (m,a_n)=1 \ , \ \forall n\geq 0$.

2018 Israel National Olympiad, 4

Tags: number theory , digit , digit sum , divisibility

The three-digit number 999 has a special property: It is divisible by 27, and its digit sum is also divisible by 27. The four-digit number 5778 also has this property, as it is divisible by 27 and its digit sum is also divisible by 27. How many four-digit numbers have this property?

2019 Tournament Of Towns, 2

Tags: number theory , sum of powers , divisibility

Consider two positive integers $a$ and $b$ such that $a^{n+1} + b^{n+1}$ is divisible by $a^n + b^n$ for infinitely many positive integers $n$. Is it necessarily true that $a = b$? (Boris Frenkin)

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$.

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]

2004 IMO Shortlist, 6

Tags: modular arithmetic , number theory , divisibility , remainder

Given an integer ${n>1}$, denote by $P_{n}$ the product of all positive integers $x$ less than $n$ and such that $n$ divides ${x^2-1}$. For each ${n>1}$, find the remainder of $P_{n}$ on division by $n$. [i]Proposed by John Murray, Ireland[/i]

2019 Czech and Slovak Olympiad III A, 3

Tags: divisibility , perfect power , number theory

Let $a,b,c,n$ be positive integers such that the following conditions hold (i) numbers $a,b,c,a+b+c$ are pairwise coprime, (ii) number $(a+b)(b+c)(c+a)(a+b+c)(ab+bc+ca)$ is a perfect $n$-th power. Prove, that the product $abc$ can be expressed as a difference of two perfect $n$-th powers.

IMSC 2024, 1

Tags: divisibility , number theory

For a positive integer $n$ denote by $P_0(n)$ the product of all non-zero digits of $n$. Let $N_0$ be the set of all positive integers $n$ such that $P_0(n)|n$. Find the largest possible value of $\ell$ such that $N_0$ contains infinitely many strings of $\ell$ consecutive integers. [i]Proposed by Navid Safaei, Iran[/i]

2023 Ukraine National Mathematical Olympiad, 9.2

Tags: number theory , divisibility

Positive integers $a_1, a_2, \ldots, a_{101}$ are such that $a_i+1$ is divisible by $a_{i+1}$ for all $1 \le i \le 101$, where $a_{102} = a_1$. What is the largest possible value of $\max(a_1, a_2, \ldots, a_{101})$? [i]Proposed by Oleksiy Masalitin[/i]

2017 Germany Team Selection Test, 3

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]

2015 Czech and Slovak Olympiad III A, 6

Tags: divisibility , number theory , combinatorics , pigeonhole principle

Integer $n>2$ is given. Find the biggest integer $d$, for which holds, that from any set $S$ consisting of $n$ integers, we can find three different (but not necesarilly disjoint) nonempty subsets, such that sum of elements of each of them is divisible by $d$.

2007 IMO Shortlist, 6

Tags: quadratic , number theory , vieta jumping , divisibility

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]