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: 715

2019 Moldova Team Selection Test, 12
Tags: number theory , prime numbers
Let $p\ge 5$ be a prime number. Prove that there exist positive integers $m$ and $n$ with $m+n\le \frac{p+1}{2}$ for which $p$ divides $2^n\cdot 3^m-1.$

1994 Bulgaria National Olympiad, 3
Tags: number theory , prime numbers
Let $p$ be a prime number, determine all positive integers $(x, y, z)$ such that: $x^p + y^p = p^z$

2021 Iran Team Selection Test, 4
Tags: number theory , polynomial , prime numbers , function , algebra , Combi
Assume $\Omega(n),\omega(n)$ be the biggest and smallest prime factors of $n$ respectively . Alireza and Amin decided to play a game. First Alireza chooses $1400$ polynomials with integer coefficients. Now Amin chooses $700$ of them, the set of polynomials of Alireza and Amin are $B,A$ respectively . Amin wins if for all $n$ we have : $$\max_{P \in A}(\Omega(P(n))) \ge \min_{P \in B}(\omega(P(n)))$$ Who has the winning strategy. Proposed by [i]Alireza Haghi[/i]

2015 AIME Problems, 3
Tags: number theory , prime numbers , AMC , AIME , AIME I , factoring polynomials
There is a prime number $p$ such that $16p+1$ is the cube of a positive integer. Find $p$.

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

2016 Bosnia and Herzegovina Team Selection Test, 3
Tags: number theory , prime numbers , Integer sequence
For an infinite sequence $a_1<a_2<a_3<...$ of positive integers we say that it is [i]nice[/i] if for every positive integer $n$ holds $a_{2n}=2a_n$. Prove the following statements: $a)$ If there is given a [i]nice[/i] sequence and prime number $p>a_1$, there exist some term of the sequence which is divisible by $p$. $b)$ For every prime number $p>2$, there exist a [i]nice[/i] sequence such that no terms of the sequence are divisible by $p$.

2018 Korea Winter Program Practice Test, 4
Tags: number theory , prime numbers
Let $p=4k+1$ be a prime. $S$ is a set of all possible residues equal or smaller then $2k$ when $\frac{1}{2} \binom{2k}{k} n^k$ is divided by $p$. Show that \[ \sum_{x \in S} x^2 =p \]

2009 Princeton University Math Competition, 3
Tags: number theory , prime numbers
Find the sum of all prime numbers $p$ which satisfy \[p = a^4 + b^4 + c^4 - 3\] for some primes (not necessarily distinct) $a$, $b$ and $c$.

2011 Dutch IMO TST, 4
Tags: recurrence relation , Sequence , prime numbers , number theory
Prove that there exists no in nite sequence of prime numbers $p_0, p_1, p_2,...$ such that for all positive integers $k$: $p_k = 2p_{k-1} + 1$ or $p_k = 2p_{k-1} - 1$.

2024 Bangladesh Mathematical Olympiad, P1
Tags: number theory , Diophantine equation , modular arithmetic , prime numbers
Find all prime numbers $p$ and $q$ such that\[p^3-3^q=10.\] [i]Proposed by Md. Fuad Al Alam[/i]

1982 Bundeswettbewerb Mathematik, 4
Tags: prime numbers , factorization , number theory
Let $n$ be a positive integer. If $4^n + 2^n + 1$ is a prime, prove that $n$ is a power of three.

2005 China Team Selection Test, 1
Tags: number theory , prime divisors , prime numbers , Divisibility , IMO Shortlist , power of 2 , Zsigmondy
Let $ b, m, n$ be positive integers such that $ b > 1$ and $ m \neq n.$ Prove that if $ b^m \minus{} 1$ and $ b^n \minus{} 1$ have the same prime divisors, then $ b \plus{} 1$ is a power of 2.

2020 Italy National Olympiad, #5
Tags: function , number theory , prime numbers
Le $S$ be the set of positive integers greater than or equal to $2$. A function $f: S\rightarrow S$ is italian if $f$ satifies all the following three conditions: 1) $f$ is surjective 2) $f$ is increasing in the prime numbers(that is, if $p_1<p_2$ are prime numbers, then $f(p_1)<f(p_2)$) 3) For every $n\in S$ the number $f(n)$ is the product of $f(p)$, where $p$ varies among all the primes which divide $n$ (For instance, $f(360)=f(2^3\cdot 3^2\cdot 5)=f(2)\cdot f(3)\cdot f(5)$). Determine the maximum and the minimum possible value of $f(2020)$, when $f$ varies among all italian functions.

1987 IMO, 3
Tags: algebra , polynomial , number theory , prime numbers , IMO , IMO 1987 , IMO Shortlist
Let $n\ge2$ be an integer. Prove that if $k^2+k+n$ is prime for all integers $k$ such that $0\le k\le\sqrt{n\over3}$, then $k^2+k+n$ is prime for all integers $k$ such that $0\le k\le n-2$.

2011 Bosnia And Herzegovina - Regional Olympiad, 2
Tags: number theory , prime numbers
If $p>2$ is prime number and $m$ and $n$ are positive integers such that $$\frac{m}{n}=1+\frac{1}{2}+\frac{1}{3}+...+\frac{1}{p-1}$$ Prove that $p$ divides $m$

2015 IFYM, Sozopol, 1
Tags: number theory , prime numbers , Divisibility
Let $p$, $q$ be two distinct prime numbers and $n$ be a natural number, such that $pq$ divides $n^{pq}+1$. Prove that, if $p^3 q^3$ divides $n^{pq}+1$, then $p^2$ or $q^2$ divides $n+1$.

1974 Putnam, A3
Tags: Putnam , number theory , Perfect Square , prime numbers
A well-known theorem asserts that a prime $p > 2$ can be written as the sum of two perfect squares ($p = m^2 +n^2$ , with $m$ and $n$ integers) if and only if $p \equiv 1$ (mod $4$). Assuming this result, find which primes $p > 2$ can be written in each of the following forms, using integers $x$ and $y$: a) $x^2 +16y^2, $ b) $4x^2 +4xy+ 5y^2.$

2023 Poland - Second Round, 6
Tags: number theory , prime numbers , combinatorics , Chessboard
Given a chessboard $n \times n$, where $n\geq 4$ and $p=n+1$ is a prime number. A set of $n$ unit squares is called [i]tactical[/i] if after putting down queens on these squares, no two queens are attacking each other. Prove that there exists a partition of the chessboard into $n-2$ tactical sets, not containing squares on the main diagonals. Queens are allowed to move horizontally, vertically and diagonally.

Russian TST 2014, P2
Tags: number theory , prime numbers
Let $p,q$ and $s{}$ be prime numbers such that $2^sq =p^y-1$ where $y > 1.$ Find all possible values of $p.$

2015 Belarus Team Selection Test, 3
Tags: IMO Shortlist , combinatorics , algebra , prime numbers
Construct a tetromino by attaching two $2 \times 1$ dominoes along their longer sides such that the midpoint of the longer side of one domino is a corner of the other domino. This construction yields two kinds of tetrominoes with opposite orientations. Let us call them $S$- and $Z$-tetrominoes, respectively. Assume that a lattice polygon $P$ can be tiled with $S$-tetrominoes. Prove that no matter how we tile $P$ using only $S$- and $Z$-tetrominoes, we always use an even number of $Z$-tetrominoes. [i]Proposed by Tamas Fleiner and Peter Pal Pach, Hungary[/i]

2021 Iran Team Selection Test, 4
Tags: number theory , polynomial , prime numbers , function , algebra , Combi
Assume $\Omega(n),\omega(n)$ be the biggest and smallest prime factors of $n$ respectively . Alireza and Amin decided to play a game. First Alireza chooses $1400$ polynomials with integer coefficients. Now Amin chooses $700$ of them, the set of polynomials of Alireza and Amin are $B,A$ respectively . Amin wins if for all $n$ we have : $$\max_{P \in A}(\Omega(P(n))) \ge \min_{P \in B}(\omega(P(n)))$$ Who has the winning strategy. Proposed by [i]Alireza Haghi[/i]

2021/2022 Tournament of Towns, P7
Tags: number theory , prime numbers , geometry
Let $p$ be a prime number and let $M$ be a convex polygon. Suppose that there are precisely $p$ ways to tile $m$ with equilateral triangles with side $1$ and squares with side $1$. Show there is some side of $M$ of length $p-1$.

2018 Baltic Way, 20
Tags: number theory , prime numbers
Find all the triples of positive integers $(a,b,c)$ for which the number \[\frac{(a+b)^4}{c}+\frac{(b+c)^4}{a}+\frac{(c+a)^4}{b}\] is an integer and $a+b+c$ is a prime.

2017 Poland - Second Round, 6
Tags: number theory , prime numbers
A prime number $p > 2$ and $x,y \in \left\{ 1,2,\ldots, \frac{p-1}{2} \right\}$ are given. Prove that if $x\left( p-x\right)y\left( p-y\right)$ is a perfect square, then $x = y$.

2009 Princeton University Math Competition, 5
Tags: number theory , prime numbers
Find the sum of all prime numbers $p$ which satisfy \[p = a^4 + b^4 + c^4 - 3\] for some primes (not necessarily distinct) $a$, $b$ and $c$.