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

2000 Tuymaada Olympiad, 5
Tags: number theory , prime , divisible
Are there prime $p$ and $q$ larger than $3$, such that $p^2-1$ is divisible by $q$ and $q^2-1$ divided by $p$?

2013 Dutch BxMO/EGMO TST, 3
Tags: prime , diophantine equation , number theory
Find all triples $(x,n,p)$ of positive integers $x$ and $n$ and primes $p$ for which the following holds $x^3 + 3x + 14 = 2 p^n$

2015 Ukraine Team Selection Test, 3
Tags: prime , number theory
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]

2019 Saudi Arabia Pre-TST + Training Tests, 5.1
Tags: prime , divide , divisible , number theory
Let $n$ be a positive integer and $p > n+1$ a prime. Prove that $p$ divides the following sum $S = 1^n + 2^n +...+ (p - 1)^n$

2017 Puerto Rico Team Selection Test, 5
Tags: prime , odd , number theory
Find a pair prime numbers $(p, q)$, $p> q$ of , if any, such that $\frac{p^2 - q^2}{4}$ is an odd integer.

2019 Saudi Arabia BMO TST, 1
Tags: divide , divisible , prime , number theory
Let $p$ be an odd prime number. a) Show that $p$ divides $n2^n + 1$ for infinitely many positive integers n. b) Find all $n$ satisfy condition above when $p = 3$

2019 Austrian Junior Regional Competition, 4
Tags: prime , divide , divisible , number theory
Let $p, q, r$ and $s$ be four prime numbers such that $$5 <p <q <r <s <p + 10.$$ Prove that the sum of the four prime numbers is divisible by $60$. (Walther Janous)

2018 All-Russian Olympiad, 1
Tags: prime number , prime , number theory , algebra
Suppose $a_1,a_2, \dots$ is an infinite strictly increasing sequence of positive integers and $p_1, p_2, \dots$ is a sequence of distinct primes such that $p_n \mid a_n$ for all $n \ge 1$. It turned out that $a_n-a_k=p_n-p_k$ for all $n,k \ge 1$. Prove that the sequence $(a_n)_n$ consists only of prime numbers.

2014 India PRMO, 1
Tags: number theory , prime , factor , maximum
A natural number $k$ is such that $k^2 < 2014 < (k +1)^2$. What is the largest prime factor of $k$?

1995 IMO, 6
Tags: counting , modular arithmetic , prime , number theory
Let $ p$ be an odd prime number. How many $ p$-element subsets $ A$ of $ \{1,2,\dots,2p\}$ are there, the sum of whose elements is divisible by $ p$?

2024 Singapore MO Open, Q5
Tags: prime , grid , prime number , combinatorics , number theory
Let $p$ be a prime number. Determine the largest possible $n$ such that the following holds: it is possible to fill an $n\times n$ table with integers $a_{ik}$ in the $i$th row and $k$th column, for $1\le i,k\le n$, such that for any quadruple $i,j,k,l$ with $1\le i<j\le n$ and $1\le k<l\le n$, the number $a_{ik}a_{jl}-a_{il}a_{jk}$ is not divisible by $p$. [i]Proposed by oneplusone[/i]

2010 Estonia Team Selection Test, 1
Tags: prime , gcd , power of prime , coprime , number theory
For arbitrary positive integers $a, b$, denote $a @ b =\frac{a-b}{gcd(a,b)}$ Let $n$ be a positive integer. Prove that the following conditions are equivalent: (i) $gcd(n, n @ m) = 1$ for every positive integer $m < n$, (ii) $n = p^k$ where $p$ is a prime number and $k$ is a non-negative integer.

1987 Polish MO Finals, 5
Tags: prime , number theory , product , min
Find the smallest $n$ such that $n^2 -n+11$ is the product of four primes (not necessarily distinct).

1989 Austrian-Polish Competition, 6
Tags: prime , number theory , sequence , perfect square
A sequence $(a_n)_{n \in N}$ of squares of nonzero integers is such that for each $n$ the difference $a_{n+1} - a_n$ is a prime or the square of a prime. Show that all such sequences are finite and determine the longest sequence.

2022 239 Open Mathematical Olympiad, 5
Tags: prime , number theory
Prove that there are infinitely many positive integers $k$ such that $k(k+1)(k+2)(k+3)$ has no prime divisor of the form $8t+5.$

2000 Chile National Olympiad, 7
Tags: prime , number theory
Consider the following equation in $x$: $$ax (x^2 + ax + 1) = b (x^2 + b + 1).$$ It is known that $a, b$ are real such that $ab <0$ and furthermore the equation has exactly two integer roots positive. Prove that under these conditions $a^2 + b^2$ is not a prime number.

2000 Singapore MO Open, 2
Tags: divisible , prime , divide , gcd , number theory
Show that $240$ divides all numbers of the form $p^4 - q^4$, where p and q are prime numbers strictly greater than $5$. Show also that $240$ is the greatest common divisor of all numbers of the form $p^4 - q^4$, with $p$ and $q$ prime numbers strictly greater than $5$.

2015 Taiwan TST Round 2, 3
Tags: prime , number theory
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 India Regional Mathematical Olympiad, 5
Tags: prime , number theory , diophantine equation , positive integer , diophantine
Determine with proof all triples $(a, b, c)$ of positive integers satisfying $\frac{1}{a}+ \frac{2}{b} +\frac{3}{c} = 1$, where $a$ is a prime number and $a \le b \le c$.

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

2025 Bulgarian Spring Mathematical Competition, 9.4
Tags: prime , divisibility , sophie germain identity , functional equation , number theory , function
Determine all functions $f: \mathbb{Z}_{>0} \to \mathbb{Z}_{>0}$ such that $f(a) + 2ab + 2f(b)$ divides $f(a)^2 + 4f(b)^2$ for any positive integers $a$ and $b$.

1999 Estonia National Olympiad, 1
Tags: prime , odd , divisible , number theory
Prove that if $p$ is an odd prime, then $p^2(p^2 -1999)$ is divisible by $6$ but not by $12$.

2003 France Team Selection Test, 3
Tags: divisor , prime , algebra , number theory
Let $p_1,p_2,\ldots,p_n$ be distinct primes greater than $3$. Show that $2^{p_1p_2\cdots p_n}+1$ has at least $4^n$ divisors.

2020 Dutch Mathematical Olympiad, 4
Tags: number theory , perfect square , prime
Determine all pairs of integers $(x, y)$ such that $2xy$ is a perfect square and $x^2 + y^2$ is a prime number.

2018 Saudi Arabia GMO TST, 2
Tags: prime , divisible , divide , number theory
Let $p$ be a prime number of the form $9k + 1$. Show that there exists an integer n such that $p | n^3 - 3n + 1$.