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: 545

2023 Belarus Team Selection Test, 2.1
Tags: prime , divisibility , number theory
Find all positive integers $n>2$ such that $$ n! \mid \prod_{ p<q\le n, p,q \, \text{primes}} (p+q)$$

2004 Unirea, 2
Tags: algebra , divisibility , arithmetic sequence
Find the arithmetic sequences of $ 5 $ integers $ n_1,n_2,n_3,n_4,n_5 $ that verify $ 5|n_1,2|n_2,11|n_3,7|n_4,17|n_5. $

2015 IFYM, Sozopol, 5
Tags: prime number , divisibility , prime divisor , number theory
Does there exist a natural number $n$ with exactly 3 different prime divisors $p$, $q$, and $r$, so that $p-1\mid n$, $qr-1\mid n$, $q-1\nmid n$, $r-1\nmid n$, and $3\nmid q+r$?

2014 India Regional Mathematical Olympiad, 6
Tags: divisibility , combinatorics , number theory
In the adjacent fi gure, can the numbers $1,2,3, 4,..., 18$ be placed, one on each line segment, such that the sum of the numbers on the three line segments meeting at each point is divisible by $3$?

2024 Singapore Senior Math Olympiad, Q4
Tags: divisibility , number theory
Suppose $p$ is a prime number and $x, y, z$ are integers satisfying $0 < x < y < z <p$. If $x^3, y^3, z^3$ have equal remainders when divided by $p$, prove that $x ^ 2 + y ^ 2 + z ^ 2$ is divisible by $x + y + z$.

1980 Bundeswettbewerb Mathematik, 1
Tags: divisibility , digit , game , number theory
Six free cells are given in a row. Players $A$ and $B$ alternately write digits from $0$ to $9$ in empty cells, with $A$ starting. When all the cells are filled, one considers the obtained six-digit number $z$. Player $B$ wins if $z$ is divisible by a given natural number $n$, and loses otherwise. For which values of $n$ not exceeding $20$ can $B$ win independently of his opponent’s moves?

1979 IMO Shortlist, 7
Tags: number theory , series summation , divisibility , prime , relatively 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 $1979$.

2018 Israel Olympic Revenge, 1
Tags: prime number , divisibility , number theory , divisor
Let $n$ be a positive integer. Prove that every prime $p > 2$ that divides $(2-\sqrt{3})^n + (2+\sqrt{3})^n$ satisfy $p=1 (mod3)$

1989 Bulgaria National Olympiad, Problem 6
Tags: divisibility , number theory
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 $x^p+y^p\ne z^p$.

1969 IMO Longlists, 54
Tags: number theory , polynomial , divisibility , modular arithmetic , algebra
$(POL 3)$ Given a polynomial $f(x)$ with integer coefficients whose value is divisible by $3$ for three integers $k, k + 1,$ and $k + 2$. Prove that $f(m)$ is divisible by $3$ for all integers $m.$

2003 IMO, 6
Tags: divisibility , polynomial , number theory
Let $p$ be a prime number. Prove that there exists a prime number $q$ such that for every integer $n$, the number $n^p-p$ is not divisible by $q$.

1990 IMO Shortlist, 20
Tags: pigeonhole principle , divisibility , digit , decimal representation , multiple , number theory
Prove that every integer $ k$ greater than 1 has a multiple that is less than $ k^4$ and can be written in the decimal system with at most four different digits.

2017 China Northern MO, 2
Tags: divisibility , number theory
Prove that there exist infinitely many integers \(n\) which satisfy \(2017^2 | 1^n + 2^n + ... + 2017^n\).

1960 IMO Shortlist, 1
Tags: divisibility , decimal representation , algebra , number theory
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 $N$.

2016 Middle European Mathematical Olympiad, 4
Tags: divisibility , function , functional equation , algebra , number theory
Find all $f : \mathbb{N} \to \mathbb{N} $ such that $f(a) + f(b)$ divides $2(a + b - 1)$ for all $a, b \in \mathbb{N}$. Remark: $\mathbb{N} = \{ 1, 2, 3, \ldots \} $ denotes the set of the positive integers.

1977 IMO Longlists, 10
Tags: additive number theory , divisibility , remainder , number theory
Let $a,b$ be two natural numbers. When we divide $a^2+b^2$ by $a+b$, we the the remainder $r$ and the quotient $q.$ Determine all pairs $(a, b)$ for which $q^2 + r = 1977.$

2016 Greece Team Selection Test, 1
Tags: sequence , divisibility , number theory
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 $\gcd (m,a_n)=1 \ , \ \forall n\geq 0$.

2025 Bulgarian Winter Tournament, 11.4
Tags: prime , sequence , divisibility , infinite prime divisors , function , algebra , polynomial
Let $A$ be a set of $2025$ non-negative integers and $f: \mathbb{Z}_{>0} \to \mathbb{Z}_{>0}$ be a function with the following two properties: 1) For every two distinct positive integers $x,y$ there exists $a\in A$, such that $x-y$ divides $f(x+a) - f(y+a)$. 2) For every positive integer $N$ there exists a positive integer $t$ such that $f(x) \neq f(y)$ whenever $x,y \in [t, t+N]$ are distinct. Prove that there are infinitely many primes $p$ such that $p$ divides $f(x)$ for some positive integer $x$.

1992 IMO Longlists, 30
Tags: divisibility , number theory
Let $P_n = (19 + 92)(19^2 +92^2) \cdots(19^n +92^n)$ for each positive integer $n$. Determine, with proof, the least positive integer $m$, if it exists, for which $P_m$ is divisible by $33^{33}.$

2008 IMO Shortlist, 2
Tags: sequence , divisibility , number theory , modular arithmetic
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 China Team Selection Test, 3
Tags: divisibility , inequalities , number theory
Let $a_1, a_2, \ldots, a_n$ be $n$ positive integers that are not divisible by each other, i.e. for any $i \neq j$, $a_i$ is not divisible by $a_j$. Show that \[ a_1+a_2+\cdots+a_n \ge 1.1n^2-2n. \] [i]Note:[/i] A proof of the inequality when $n$ is sufficient large will be awarded points depending on your results.

2015 Cono Sur Olympiad, 1
Tags: divisibility , number theory
Show that, for any integer $n$, the number $n^3 - 9n + 27$ is not divisible by $81$.

2018 USA TSTST, 8
Tags: divisibility , number theory
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]

2010 Germany Team Selection Test, 3
Tags: sequence , divisibility , modular arithmetic , number theory
Find all positive integers $n$ such that there exists a sequence of positive integers $a_1$, $a_2$,$\ldots$, $a_n$ satisfying: \[a_{k+1}=\frac{a_k^2+1}{a_{k-1}+1}-1\] for every $k$ with $2\leq k\leq n-1$. [i]Proposed by North Korea[/i]

1979 IMO, 1
Tags: series summation , divisibility , prime , relatively prime , number theory
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 $1979$.