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: 364

2025 Kosovo National Mathematical Olympiad`, P4
Tags: prime , gcd , solve in natural , functional equation , number theory
Find all functions $f : \mathbb{N} \rightarrow \mathbb{N}$ for which these two conditions hold simultaneously (i) For all $m,n \in \mathbb{N}$ we have: $$ \frac{f(mn)}{\gcd(m,n)} = \frac{f(m)f(n)}{f(\gcd(m,n))};$$ (ii) For all prime numbers $p$, there exists a prime number $q$ such that $f(p^{2025})=q^{2025}$.

2013 Switzerland - Final Round, 2
Tags: prime , number theory , inequalities
Let $n$ be a natural number and $p_1, ..., p_n$ distinct prime numbers. Show that $$p_1^2 + p_2^2 + ... + p_n^2 > n^3$$

2022 IMO Shortlist, N2
Tags: prime , number theory , divisibility
Find all positive integers $n>2$ such that $$ n! \mid \prod_{ p<q\le n, p,q \, \text{primes}} (p+q)$$

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$.

2008 Tournament Of Towns, 5
Tags: combinatorics , prime , integer
The positive integers are arranged in a row in some order, each occuring exactly once. Does there always exist an adjacent block of at least two numbers somewhere in this row such that the sum of the numbers in the block is a prime number?

2017 Peru IMO TST, 10
Tags: polynomial , prime , number theory , squarefree , integer
Let $P (n)$ and $Q (n)$ be two polynomials (not constant) whose coefficients are integers not negative. For each positive integer $n$, define $x_n = 2016^{P (n)} + Q (n)$. Prove that there exist infinite primes $p$ for which there is a positive integer $m$, squarefree, such that $p | x_m$. Clarification: A positive integer is squarefree if it is not divisible by the square of any prime number.

2025 Nordic, 2
Tags: number theory , nt , prime
Let $p$ be a prime and suppose $2^{2p} \equiv 1 (\text{mod}$ $ 2p+1)$ is prime. Prove that $2p+1$ is prime$^{1}$ [size=75]$^{1}$This is a special case of Pocklington's theorem. A proof of this special case is required.[/size]

2021 Nigerian MO Round 3, Problem 3
Tags: prime , number theory , diophantine equation
Find all pairs of natural numbers $(p, n)$ with $p$ prime such that $p^6+p^5+n^3+n=n^5+n^2$.

2015 May Olympiad, 4
Tags: number theory , combinatorics , prime
The first $510$ positive integers are written on a blackboard: $1, 2, 3, ..., 510$. An [i]operation [/i] consists of of erasing two numbers whose sum is a prime number. What is the maximum number of operations in a row what can be done? Show how it is accomplished and explain why it can be done in no more operations.

2013 IMAR Test, 1
Tags: prime , number theory , fermat's little theorem
Given a prime $p \geq 5$ , show that there exist at least two distinct primes $q$ and $r$ in the range $2, 3, \ldots p-2$ such that $q^{p-1} \not\equiv 1 \pmod{p^2}$ and $r^{p-1} \not\equiv 1 \pmod{p^2}$.

2018 Ukraine Team Selection Test, 7
Tags: prime , perfect square , divisor , number theory
The prime number $p > 2$ and the integer $n$ are given. Prove that the number $pn^2$ has no more than one divisor $d$ for which $n^2+d$ is the square of the natural number. .

2022 Bulgarian Spring Math Competition, Problem 10.4
Tags: system of equations , algebra , prime
Find the smallest odd prime $p$, such that there exist coprime positive integers $k$ and $\ell$ which satisfy \[4k-3\ell=12\quad \text{ and }\quad \ell^2+\ell k +k^2\equiv 3\text{ }(\text{mod }p)\]

2021 Durer Math Competition Finals, 1
Tags: prime , number theory
Show that if the difference of two positive cube numbers is a positive prime, then this prime number has remainder $1$ after division by $6$.

2000 Singapore Team Selection Test, 2
Tags: number theory , prime , perfect square
Find all prime numbers $p$ such that $5^p + 12^p$ is a perfect square

2009 Bosnia and Herzegovina Junior BMO TST, 3
Tags: prime , number theory , division , prime number
Let $p$ be a prime number, $p\neq 3$ and let $a$ and $b$ be positive integers such that $p \mid a+b$ and $p^2\mid a^3+b^3$. Show that $p^2 \mid a+b$ or $p^3 \mid a^3+b^3$

2012 Singapore Junior Math Olympiad, 5
Tags: number theory , prime number , coprime , prime
Suppose $S = \{a_1, a_2,..., a_{15}\}$ is a set of $1 5$ distinct positive integers chosen from $2 , 3, ... , 2012$ such that every two of them are coprime. Prove that $S$ contains a prime number. (Note: Two positive integers $m, n$ are coprime if their only common factor is 1)

1998 Switzerland Team Selection Test, 6
Tags: number theory , prime , divisor
Find all prime numbers $p$ for which $p^2 +11$ has exactly six positive divisors.

1994 Abels Math Contest (Norwegian MO), 2a
Tags: number theory , diophantine , diophantine equation , prime
Find all primes $p,q,r$ and natural numbers $n$ such that $\frac{1}{p}+\frac{1}{q}+\frac{1}{r}=\frac{1}{n}$.

1965 Poland - Second Round, 4
Tags: prime , number theory
Find all prime numbers $ p $ such that $ 4p^2 + 1 $ and $ 6p^2 + 1 $ are also prime numbers.

2014 Junior Balkan Team Selection Tests - Romania, 3
Tags: combinatorics , prime , sum , number theory
Let $n \ge 5$ be an integer. Prove that $n$ is prime if and only if for any representation of $n$ as a sum of four positive integers $n = a + b + c + d$, it is true that $ab \ne cd$.

2010 Saudi Arabia Pre-TST, 2.2
Tags: prime , consecutive , sum of squares , number theory
Find all $n$ for which there are $n$ consecutive integers whose sum of squares is a prime.

2015 Belarus Team Selection Test, 1
Tags: prime , number theory
Find all positive integers $n$ such that $n=q(q^2-q-1)=r(2r+1)$ for some primes $q$ and $r$. B.Gilevich

2019 Centers of Excellency of Suceava, 2
Tags: number theory , prime , gcd
Let $ \left( s_n \right)_{n\ge 1 } $ be a sequence with $ s_1 $ and defined recursively as $ s_{n+1}=s_n^2-s_n+1. $ Prove that any two terms of this sequence are coprime. [i]Dan Nedeianu[/i]

2005 Korea Junior Math Olympiad, 6
Tags: prime , set theory , number theory
For two different prime numbers $p, q$, defi ne $S_{p,q} = \{p,q,pq\}$. If two elements in $S_{p,q}$ are numbers in the form of $x^2 + 2005y^2, (x, y \in Z)$, prove that all three elements in $S_{p,q}$ are in such form.

2021 Indonesia TST, C
Tags: counting , combinatorics , subset , prime , summation
Let $p$ be an odd prime. Determine the number of nonempty subsets from $\{1, 2, \dots, p - 1\}$ for which the sum of its elements is divisible by $p$.