Olympiad problems

2007 IMO Shortlist, 4

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]

2005 IMO Shortlist, 4

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]

2011 Belarus Team Selection Test, 3

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

Let $a, b$ be integers, and let $P(x) = ax^3+bx.$ For any positive integer $n$ we say that the pair $(a,b)$ is $n$-good if $n | P(m)-P(k)$ implies $n | m - k$ for all integers $m, k.$ We say that $(a,b)$ is $very \ good$ if $(a,b)$ is $n$-good for infinitely many positive integers $n.$ [list][*][b](a)[/b] Find a pair $(a,b)$ which is 51-good, but not very good. [*][b](b)[/b] Show that all 2010-good pairs are very good.[/list] [i]Proposed by Okan Tekman, Turkey[/i]

2016 Iran Team Selection Test, 1

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.

1989 Bulgaria National Olympiad, Problem 6

Tags: number theory , Divisibility

Let $x,y,z$ be pairwise coprime positive integers and $p\ge5$ and $q$ be prime numbers which satisfy the following conditions: (i) $6p$ does not divide $q-1$; (ii) $q$ divides $x^2+xy+y^2$; (iii) $q$ does not divide $x+y-z$. Prove that $x^p+y^p\ne z^p$.

2014 India Regional Mathematical Olympiad, 3

Tags: number theory , Divisibility , positive integers

Find all pairs of $(x, y)$ of positive integers such that $2x + 7y$ divides $7x + 2y$.

Russian TST 2014, P1

Tags: number theory , Divisibility , prime numbers

Let $p{}$ be a prime number and $x_1,x_2,\ldots,x_p$ be integers for which $x_1^n+x_2^n+\cdots+x_p^n$ is divisible by $p{}$ for any positive integer $n{}$. Prove that $x_1-x_2$ is divisible by $p{}.$

1960 IMO Shortlist, 1

Tags: algebra , number theory , Divisibility , decimal representation , IMO , IMO 1960 , IMO Shortlist

Determine all three-digit numbers $N$ having the property that $N$ is divisible by 11, and $\dfrac{N}{11}$ is equal to the sum of the squares of the digits of $N$.

1969 IMO Longlists, 49

Tags: number theory , Divisibility , combinatorics , invariant , IMO Shortlist , IMO Longlist

$(NET 4)$ A boy has a set of trains and pieces of railroad track. Each piece is a quarter of circle, and by concatenating these pieces, the boy obtained a closed railway. The railway does not intersect itself. In passing through this railway, the train sometimes goes in the clockwise direction, and sometimes in the opposite direction. Prove that the train passes an even number of times through the pieces in the clockwise direction and an even number of times in the counterclockwise direction. Also, prove that the number of pieces is divisible by $4.$

2021 Bolivian Cono Sur TST, 1

Tags: Pre TST , TST , Bolivia , Cono Sur TST , number theory , Divisibility

Find the sum of all positive integers $n$ such that $$\frac{n+11}{\sqrt{n-1}}$$ is an integer.

2015 Bosnia Herzegovina Team Selection Test, 3

Tags: composite numbers , number theory , infinitely many solutions , Divisibility

Prove that there exist infinitely many composite positive integers $n$ such that $n$ divides $3^{n-1}-2^{n-1}$.

Russian TST 2015, P1

Tags: number theory , Divisibility

Find all pairs of natural numbers $(a,b)$ satisfying the following conditions: [list] [*]$b-1$ is divisible by $a+1$ and [*]$a^2+a+2$ is divisible by $b$. [/list]

2002 Federal Math Competition of S&M, Problem 3

Tags: number theory , Divisibility

Let $m$ and $n$ be positive integers. Prove that the number $2n-1$ is divisible by $(2^m-1)^2$ if and only if $n$ is divisible by $m(2^m-1)$.

2007 Bulgarian Autumn Math Competition, Problem 9.4

Tags: number theory , Divisibility

Find the smallest natural number, which divides $2^{n}+15$ for some natural number $n$ and can be expressed in the form $3x^2-4xy+3y^2$ for some integers $x$ and $y$.

1946 Putnam, B5

Tags: Putnam , ceiling function , Divisibility

Show that $\lceil (\sqrt{3}+1)^{2n})\rceil$ is divisible by $2^{n+1}.$

1999 IMO Shortlist, 7

Tags: counting , Subsets , Set systems , Divisibility , combinatorics , IMO Shortlist , combinatorial inequality

Let $p >3$ be a prime number. For each nonempty subset $T$ of $\{0,1,2,3, \ldots , p-1\}$, let $E(T)$ be the set of all $(p-1)$-tuples $(x_1, \ldots ,x_{p-1} )$, where each $x_i \in T$ and $x_1+2x_2+ \ldots + (p-1)x_{p-1}$ is divisible by $p$ and let $|E(T)|$ denote the number of elements in $E(T)$. Prove that \[|E(\{0,1,3\})| \geq |E(\{0,1,2\})|\] with equality if and only if $p = 5$.

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

1992 Spain Mathematical Olympiad, 1

Tags: Spain , number theory , Divisibility , combinatorics , congruence

Determine the smallest number N, multiple of 83, such that N^2 has 63 positive divisors.

2023 4th Memorial "Aleksandar Blazhevski-Cane", P6

Tags: number theory , functions , Divisibility , positive integers

Denote by $\mathbb{N}$ the set of positive integers. Find all functions $f:\mathbb{N} \rightarrow \mathbb{N}$ such that: [b]•[/b] For all positive integers $a> 2023^{2023}$ it holds that $f(a) \leq a$. [b]•[/b] $\frac{a^2f(b)+b^2f(a)}{f(a)+f(b)}$ is a positive integer for all $a,b \in \mathbb{N}$. [i]Proposed by Nikola Velov[/i]

2012 IMO Shortlist, N8

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

Prove that for every prime $p>100$ and every integer $r$, there exist two integers $a$ and $b$ such that $p$ divides $a^2+b^5-r$.

2018 USA TSTST, 8

Tags: number theory , USA , USA TSTST , Divisibility

For which positive integers $b > 2$ do there exist infinitely many positive integers $n$ such that $n^2$ divides $b^n+1$? [i]Evan Chen and Ankan Bhattacharya[/i]

2019 OMMock - Mexico National Olympiad Mock Exam, 4

Tags: number theory , Divisibility

Find all positive integers $n$ such that it is possible to split the numbers from $1$ to $2n$ in two groups $(a_1,a_2,..,a_n)$, $(b_1,b_2,...,b_n)$ in such a way that $2n\mid a_1a_2\cdots a_n+b_1b_2\cdots b_n-1$. [i]Proposed by Alef Pineda[/i]

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 $x$.)

2016 Irish Math Olympiad, 1

Tags: number theory , Divisibility

If the three-digit number $ABC$ is divisible by $27$, prove that the three-digit numbers $BCA$ and $CAB$ are also divisible by $27$.

1969 IMO Longlists, 23

Tags: number theory , relatively prime , Divisibility , Eulers function , IMO Shortlist , IMO Longlist

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