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

2022 Chile National Olympiad, 3
Tags: number theory , prime , digit
The $19$ numbers $472$ , $473$ , $...$ , $490$ are juxtaposed in some order to form a $57$-digit number. Can any of the numbers thus obtained be prime?

2023 Switzerland - Final Round, 6
Tags: number theory , prime , divisibility
Find all positive integers $n>2$ such that $$ n! \mid \prod_{ p<q\le n, p,q \, \text{primes}} (p+q)$$

2022 Ecuador NMO (OMEC), 6
Tags: number theory , prime
Prove that for all prime $p \ge 5$, there exist an odd prime $q \not= p$ such that $q$ divides $(p-1)^p + 1$

2018 Chile National Olympiad, 1
Tags: number theory , prime number , prime
Is it possible to choose five different positive integers so that the sum of any three of them is a prime number?

1999 Romania National Olympiad, 2
Tags: number theory , prime
Let $a, b, c$ be non zero integers,$ a\ne c$ such that $$\frac{a}{c}=\frac{a^2+b^2}{c^2+b^2}$$ Prove that $a^2 +b^2 +c^2$ cannot be a prime number.

2018 Saudi Arabia IMO TST, 1
Tags: theory , divisor , prime , prime divisor
Denote $S$ as the set of prime divisors of all integers of form $2^{n^2+1} - 3^n, n \in Z^+$. Prove that $S$ and $P-S$ both contain infinitely many elements (where $P$ is set of prime numbers).

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

2015 Brazil Team Selection Test, 2
Tags: number theory , prime
Find all triples $(p, x, y)$ consisting of a prime number $p$ and two positive integers $x$ and $y$ such that $x^{p -1} + y$ and $x + y^ {p -1}$ are both powers of $p$. [i]Proposed by Belgium[/i]

2012 Dutch IMO TST, 1
Tags: number theory , prime , gcd , greatest common divisor , perfect power
For all positive integers $a$ and $b$, we de ne $a @ b = \frac{a - b}{gcd(a, b)}$ . Show that for every integer $n > 1$, the following holds: $n$ is a prime power if and only if for all positive integers $m$ such that $m < n$, it holds that $gcd(n, n @m) = 1$.

2007 Korea Junior Math Olympiad, 8
Tags: prime , positive integer , divisor , number theory
Prime $p$ is called [i]Prime of the Year[/i] if there exists a positive integer $n$ such that $n^2+ 1 \equiv 0$ ($mod p^{2007}$). Prove that there are infi nite number of [i]Primes of the Year[/i].

2013 IMAR Test, 1
Tags: fermat's little theorem , number theory , prime
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 $r^{p-1} \not\equiv 1 \pmod{p^2}$.

1958 Poland - Second Round, 1
Tags: prime , number theory , prime number
Prove that if $ a $ is an integer different from $ 1 $ and $ - 1 $, then $ a^4 + 4 $ is not a prime number.

2021 Nordic, 1
Tags: number theory , prime
On a blackboard a finite number of integers greater than one are written. Every minute, Nordi additionally writes on the blackboard the smallest positive integer greater than every other integer on the blackboard and not divisible by any of the numbers on the blackboard. Show that from some point onwards Nordi only writes primes on the blackboard.

2008 Singapore Junior Math Olympiad, 4
Tags: sum , prime , number theory
Six distinct positive integers $a,b,c.d,e, f$ are given. Jack and Jill calculated the sums of each pair of these numbers. Jack claims that he has $10$ prime numbers while Jill claims that she has $9$ prime numbers among the sums. Who has the correct claim?

2015 Ukraine Team Selection Test, 3
Tags: number theory , prime
Find all triples $(p, x, y)$ consisting of a prime number $p$ and two positive integers $x$ and $y$ such that $x^{p -1} + y$ and $x + y^ {p -1}$ are both powers of $p$. [i]Proposed by Belgium[/i]

1997 Estonia National Olympiad, 1
Tags: number theory , composite , prime
Prove that a positive integer $n$ is composite if and only if there exist positive integers $a,b,x,y$ such that $a+b = n$ and $\frac{x}{a}+\frac{y}{b}= 1$.

1995 India National Olympiad, 6
Tags: greatest common divisor , prime , number theory
Find all primes $p$ for which the quotient \[ \dfrac{2^{p-1} - 1 }{p} \] is a square.

2022 Bulgarian Spring Math Competition, Problem 9.3
Tags: algebra , system of equations , prime
Find all primes $p$, such that there exist positive integers $x$, $y$ which satisfy $$\begin{cases} p + 49 = 2x^2\\ p^2 + 49 = 2y^2\\ \end{cases}$$

2017 Israel Oral Olympiad, 3
Tags: number theory , prime , prime number
2017 prime numbers $p_1,...,p_{2017}$ are given. Prove that $\prod_{i<j} (p_i^{p_j}-p_j^{p_i})$ is divisible by 5777.

2009 Bosnia and Herzegovina Junior BMO TST, 3
Tags: number theory , prime , 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 $p^3 \mid a^3+b^3$

Oliforum Contest V 2017, 3
Tags: number theory , product , prime
Do there exist (not necessarily distinct) primes $p_1,..., p_k$ and $q_1,...,q_n$ such that $$p_1! \cdot \cdot \cdot p_k! \cdot 2017 = q_1! \cdot \cdot \cdot q_n! \cdot 2016 \,\,?$$ (Paolo Leonetti)

2011 Indonesia TST, 4
Tags: number theory , prime , sum of divisors , divide
Given $N = 2^ap_1p_2...p_m$, $m \ge 1$, $a \in N$ with $p_1, p_2,..., p_m$ are different primes. It is known that $\sigma (N) = 3N $ where $\sigma (N)$ is the sum of all positive integers which are factors of $N$. Show that there exists a prime number $p$ such that $2^p- 1$ is also a prime, and $2^p - 1|N$.

2025 VJIMC, 1
Tags: number theory , divisibility , prime
Let $a\geq 2$ be an integer. Prove that there exists a positive integer $b$ with the following property: For each positive integer $n$, there is a prime number $p$ (possibly depending on $a,b,n$) such that $a^n + b$ is divisible by $p$, but not divisible by $p^2$.

2014 IFYM, Sozopol, 2
Tags: number theory , point , circles , prime
The radius $r$ of a circle with center at the origin is an odd integer. There is a point ($p^m, q^n$) on the circle, with $p,q$ prime numbers and $m,n$ positive integers. Determine $r$.

2013 Abels Math Contest (Norwegian MO) Final, 3
Tags: number theory , prime , divide
A prime number $p \ge 5$ is given. Write $\frac13+\frac24+... +\frac{p -3}{p - 1}=\frac{a}{b}$ for natural numbers $a$ and $b$. Show that $p$ divides $a$.