This website contains problems from math contests. Problems and corresponding tags were obtained from the Art of Problem Solving website.

Tags were heavily modified to better represent problems.

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Found problems: 387

2022 IFYM, Sozopol, 5
Tags: algebra , divide
Find all functions $f : N \to N$ such that $f(p)$ divides $f(n)^p -n$ by any natural number $n$ and prime number $p$.

1949-56 Chisinau City MO, 7
Tags: factorial , number theory , divisible , prime , divide
Prove that if the product $1\cdot 2\cdot ...\cdot n$ ($n> 3$) is not divisible by $n + 1$, then $n + 1$ is prime.

2016 Saudi Arabia GMO TST, 3
Tags: divide , divisor , polynomial , algebra
Find all polynomials $P,Q \in Z[x]$ such that every positive integer is a divisor of a certain nonzero term of the sequence $(x_n)_{n=0}^{\infty}$ given by the conditions: $x_0 = 2016$, $x_{2n+1} = P(x_{2n})$, $x_{2n+2} = Q(x_{2n+1})$ for all $n \ge 0$

2010 Thailand Mathematical Olympiad, 6
Tags: divide , number theory , prime
Show that no triples of primes $p, q, r$ satisfy $p > r, q > r$, and $pq | r^p + r^q$

2013 QEDMO 13th or 12th, 2
Tags: binomial , number theory , divisible , divide
Let $p$ be a prime number and $n, k$ and $q$ natural numbers, where $q\le \frac{n -1}{p-1}$ should be. Let $M$ be the set of all integers $m$ from $0$ to $n$, for which $m-k$ is divisible by $p$. Show that $$\sum_{m \in M} (-1) ^m {n \choose m}$$ is divisible by $p^q$.

2012 Thailand Mathematical Olympiad, 7
Tags: number theory , greatest common divisor , gcd , divide
Let $a, b, m$ be integers such that gcd $(a, b) = 1$ and $5 | ma^2 + b^2$ . Show that there exists an integer $n$ such that $5 | m - n^2$.

2020 Malaysia IMONST 2, 3
Tags: number theory , divisible , divide
Given integers $a$ and $b$ such that $a^2+b^2$ is divisible by $11$. Prove that $a$ and $b$ are both divisible by $11$.

2021 Austrian MO Regional Competition, 4
Tags: number theory , divide , divisible
Determine all triples $(x, y, z)$ of positive integers satisfying $x | (y + 1)$, $y | (z + 1)$ and $z | (x + 1)$. (Walther Janous)

2018 Saudi Arabia BMO TST, 3
Tags: divide , divisor , number theory
Find all positive integers $n$ such that $\phi (n)$ is a divisor of $n^2+3$.

2020 Kazakhstan National Olympiad, 1
Tags: divide , number theory
Find all pairs $ (m, n) $ of natural numbers such that $ n ^ 4 \ | \ 2m ^ 5 - 1 $ and $ m ^ 4 \ | \ 2n ^ 5 + 1 $.

2016 Czech-Polish-Slovak Junior Match, 3
Tags: combinatorial geometry , divide , geometry , 3d geometry , prism , combinatorics
Find all integers $n \ge 3$ with the following property: it is possible to assign pairwise different positive integers to the vertices of an $n$-gonal prism in such a way that vertices with labels $a$ and $b$ are connected by an edge if and only if $a | b$ or $b | a$. Poland

2010 Cuba MO, 5
Tags: number theory , divide , divisible
Let $p\ge 2$ be a prime number and $a\ge 1$ be an integer different from $p$. Find all pairs $(a, p)$ such that $a + p | a^2 + p^2$.

2003 Austria Beginners' Competition, 3
Tags: number theory , consecutive , divide , divisible
a) Show that the product of $5$ consecutive even integers is divisible by $15$. b) Determine the largest integer $D$ such that the product of $5$ consecutive even integers is always divisible by $D$.

2015 Balkan MO Shortlist, N5
Tags: number theory , maximum , power of 2 , divide , product
For a positive integer $s$, denote with $v_2(s)$ the maximum power of $2$ that divides $s$. Prove that for any positive integer $m$ that: $$v_2\left(\prod_{n=1}^{2^m}\binom{2n}{n}\right)=m2^{m-1}+1.$$ (FYROM)

2015 Dutch BxMO/EGMO TST, 1
Tags: number theory , divisor , divide
Let $m$ and $n$ be positive integers such that $5m+ n$ is a divisor of $5n +m$. Prove that $m$ is a divisor of $n$.

1955 Kurschak Competition, 2
Tags: number theory , divide , digit
How many five digit numbers are divisible by $3$ and contain the digit $6$?

1974 Dutch Mathematical Olympiad, 2
Tags: number theory , divide , divisible
$n>2$ numbers, $ x_1, x_2, ..., x_n$ are odd . Prove that $4$ divides $$ x_1x_2+x_2x_3+...+x_{n-1}x_n+x_nx_1 -n.$$

2013 Saudi Arabia BMO TST, 4
Tags: divide , divisible , number theory
Find all positive integers $n < 589$ for which $589$ divides $n^2 + n + 1$.

2004 Denmark MO - Mohr Contest, 2
Tags: number theory , divide , divisible
Show that if $a$ and $b$ are integer numbers, and $a^2 + b^2 + 9ab$ is divisible by $11$, then $a^2-b^2$ divisible by $11$.

2018 Rioplatense Mathematical Olympiad, Level 3, 3
Tags: number theory , coprime , coprime numbers , sum of powers , divide
Determine all the triples $\{a, b, c \}$ of positive integers coprime (not necessarily pairwise prime) such that $a + b + c$ simultaneously divides the three numbers $a^{12} + b^{12}+ c^{12}$, $ a^{23} + b^{23} + c^{23} $ and $ a^{11004} + b^{11004} + c^{11004}$

2007 Postal Coaching, 6
Tags: digit , number theory , divide , divisible
Consider all the $7$-digit numbers formed by the digits $1,2 , 3,...,7$ each digit being used exactly once in all the $7! $ numbers. Prove that no two of them have the property that one divides the other.

1994 Tournament Of Towns, (417) 5
Tags: number theory , digit , divisible , divide
Find the maximal integer $ M$ with nonzero last digit (in its decimal representation) such that after crossing out one of its digits (not the first one) we can get an integer that divides $M$. (A Galochkin)

2003 Singapore Senior Math Olympiad, 1
Tags: perfect square , divisible , divide , number theory
It is given that n is a positive integer such that both numbers $2n + 1$ and $3n + 1$ are complete squares. Is it true that $n$ must be divisible by $40$ ? Justify your answer.

2014 Czech-Polish-Slovak Junior Match, 4
Tags: number theory , divide , divisible , digit
The number $a_n$ is formed by writing in succession, without spaces, the numbers $1, 2, ..., n$ (for example, $a_{11} = 1234567891011$). Find the smallest number t such that $11 | a_t$.

2017 Saudi Arabia JBMO TST, 2
Tags: divide , number theory
A positive integer $k > 1$ is called nice if for any pair $(m, n)$ of positive integers satisfying the condition $kn + m | km + n$ we have $n | m$. 1. Prove that $5$ is a nice number. 2. Find all the nice numbers.