Olympiad problems

KoMaL A Problems 2020/2021, A. 787

Let $p_n$ denote the $n^{\text{th}}$ prime number and define $a_n=\lfloor p_n\nu\rfloor$ for all positive integers $n$ where $\nu$ is a positive irrational number. Is it possible that there exist only finitely many $k$ such that $\binom{2a_k}{a_k}$ is divisible by $p_i^{10}$ for all $i=1,2,\ldots,2020?$ [i]Proposed by Superguy and ayan.nmath[/i]

2015 Indonesia MO, 5

Tags: divisibility , number theory

Given positive integers $a,b,c,d$ such that $a\mid c^d$ and $b\mid d^c$. Prove that \[ ab\mid (cd)^{max(a,b)} \]

2005 Federal Math Competition of S&M, Problem 1

Tags: power mean , divisibility , algebra , number theory

Let $a$ and $b$ be positive integers and $K=\sqrt{\frac{a^2+b^2}2}$, $A=\frac{a+b}2$. If $\frac KA$ is a positive integer, prove that $a=b$.

2018-IMOC, N3

Tags: number theory , divisibility

Find all pairs of positive integers $(x,y)$ so that $$\frac{(x^2-x+1)(y^2-y+1)}{xy}\in\mathbb N.$$

1969 IMO Shortlist, 24

Tags: algebra , polynomial , number theory , divisibility

$(GBR 1)$ The polynomial $P(x) = a_0x^k + a_1x^{k-1} + \cdots + a_k$, where $a_0,\cdots, a_k$ are integers, is said to be divisible by an integer $m$ if $P(x)$ is a multiple of $m$ for every integral value of $x$. Show that if $P(x)$ is divisible by $m$, then $a_0 \cdot k!$ is a multiple of $m$. Also prove that if $a, k,m$ are positive integers such that $ak!$ is a multiple of $m$, then a polynomial $P(x)$ with leading term $ax^k$can be found that is divisible by $m.$

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 $1979$.

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]

1989 IMO Shortlist, 27

Tags: modular arithmetic , number theory , least common multiple , divisibility

Let $ m$ be a positive odd integer, $ m > 2.$ Find the smallest positive integer $ n$ such that $ 2^{1989}$ divides $ m^n \minus{} 1.$

2001 Mongolian Mathematical Olympiad, Problem 3

Tags: number theory , divisibility

Let $a,b$ be coprime positive integers with $a$ even and $a>b$. Show that there exist infinitely many pairs $(m,n)$ of coprime positive integers such that $m\mid a^{n-1}-b^{n-1}$ and $n\mid a^{m-1}-b^{m-1}$.

2007 Nicolae Coculescu, 4

Tags: totient , euler , number theory , divisibility

Prove that $ p $ divides $ \varphi (1+a^p) , $ where $ a\ge 2 $ is a natural number, $ p $ is a prime, and $ \varphi $ is Euler's totient. [i]Cristinel Mortici[/i]

2015 IFYM, Sozopol, 5

Tags: number theory , divisibility , prime

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 $p$.

2022 Switzerland Team Selection Test, 1

Tags: number theory , divisibility , number base

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]

2007 Gheorghe Vranceanu, 1

Tags: induction , algebra , number theory , divisibility

Let $ \left( x_n\right)_{n\ge 1} $ be a sequence of integers defined recursively as $ x_{n+2}=5x_{n+1}-x_n. $ Prove that $ \left( x_n\right)_{n\ge 1} $ has a subsequence whose terms are multiples of $ 22 $ if $ \left( x_n\right)_{n\ge 1} $ has a term that is multiple of $ 22. $

1979 IMO Longlists, 25

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 $1979$.

2019 OMMock - Mexico National Olympiad Mock Exam, 2

Tags: number theory , divisibility , algebra

Find all pairs of positive integers $(m, n)$ such that $m^2-mn+n^2+1$ divides both numbers $3^{m+n}+(m+n)!$ and $3^{m^3+n^3}+m+n$. [i]Proposed by Dorlir Ahmeti[/i]

1999 IMO, 4

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$.

1975 IMO, 4

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.)

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]

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 $n$.

2001 Slovenia National Olympiad, Problem 1

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

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 $5$.

2014 German National Olympiad, 2

Tags: combinatorics , number theory , divisibility , sequence , formula

For a positive integer $n$, let $y_n$ be the number of $n$-digit positive integers containing only the digits $2,3,5, 7$ and which do not have a $5$ directly to the right of a $2.$ If $r\geq 1$ and $m\geq 2$ are integers, prove that $y_{m-1}$ divides $y_{rm-1}.$