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: 916

2015 NZMOC Camp Selection Problems, 4
Tags: diophantine , diophantine equation , number theory
For which positive integers $m$ does the equation: $$(ab)^{2015} = (a^2 + b^2)^m$$ have positive integer solutions?

2007 QEDMO 4th, 1
Tags: diophantine equation , diophantine , number theory
Find all primes $p,$ $q,$ $r$ satisfying $p^{2}+2q^{2}=r^{2}.$

2009 Korea Junior Math Olympiad, 8
Tags: prime number , diophantine equation , number theory
Let a, b, c, d, and e be positive integers. Are there any solutions to $a^2+b^3+c^5+d^7=e^{11}$?

1997 Brazil Team Selection Test, Problem 2
Tags: number theory , diophantine equation
We say that a subset $A$ of $\mathbb N$ is good if for some positive integer $n$, the equation $x-y=n$ admits infinitely many solutions with $x,y\in A$. If $A_1,A_2,\ldots,A_{100}$ are sets whose union is $\mathbb N$, prove that at least one of the $A_i$s is good.

2018 Pan-African Shortlist, N1
Tags: number theory , diophantine equation
Does there exist positive integers $a, b, c$ such that $4(ab - a - c^2) = b$?

1982 Tournament Of Towns, (025) 3
Tags: diophantine equation , factorial , diophantine , number theory
Prove that the equation $m!n! = k!$ has infinitely many solutions in which $m, n$ and $k$ are natural numbers greater than unity .

2010 Saudi Arabia BMO TST, 4
Tags: diophantine , diophantine equation , number theory
Find all primes $p, q$ satisfying the equation $2p^q - q^p = 7.$

2008 Greece JBMO TST, 4
Tags: diophantine equation , product , sum , integer , number theory
Product of two integers is $1$ less than three times of their sum. Find those integers.

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 $x^3 + 3x + 14 = 2 p^n$

1999 Abels Math Contest (Norwegian MO), 2a
Tags: diophantine equation , number theory , diophantine
Find all integers $m$ and $n$ such that $2m^2 +n^2 = 2mn+3n$

2022 Nigerian Senior MO Round 2, Problem 6
Tags: diophantine equation , number theory
Let $k, l, m, n$ be positive integers. Given that $k+l+m+n=km=ln$, find all possible values of $k+l+m+n$.

2021 Canadian Mathematical Olympiad Qualification, 6
Tags: diophantine equation , algebra
Show that $(w, x, y, z)=(0,0,0,0)$ is the only integer solution to the equation $$w^{2}+11 x^{2}-8 y^{2}-12 y z-10 z^{2}=0$$

1989 IMO Shortlist, 31
Tags: diophantine equation , linear equation , counting , algebra
Let $ a_1 \geq a_2 \geq a_3 \in \mathbb{Z}^\plus{}$ be given and let N$ (a_1, a_2, a_3)$ be the number of solutions $ (x_1, x_2, x_3)$ of the equation \[ \sum^3_{k\equal{}1} \frac{a_k}{x_k} \equal{} 1.\] where $ x_1, x_2,$ and $ x_3$ are positive integers. Prove that \[ N(a_1, a_2, a_3) \leq 6 a_1 a_2 (3 \plus{} ln(2 a_1)).\]

1973 Chisinau City MO, 70
Tags: diophantine , diophantine equation , number theory
The natural numbers $p, q$ satisfy the relation $p^p + q^q = p^q + q^p$. Prove that $p = q$.

2000 Swedish Mathematical Competition, 3
Tags: diophantine , diophantine equation , number theory
Are there any integral solutions to $n^2 + (n+1)^2 + (n+2)^2 = m^2$ ?

PEN H Problems, 80
Tags: diophantine equation
Prove that if $a, b, c, d$ are integers such that $d=( a+\sqrt[3]{2}b+\sqrt[3]{4}c)^{2}$ then $d$ is a perfect square.

PEN H Problems, 75
Tags: diophantine equation
Let $a,b$, and $x$ be positive integers such that $x^{a+b}=a^b{b}$. Prove that $a=x$ and $b=x^{x}$.

PEN H Problems, 37
Tags: diophantine equation , modular arithmetic
Prove that for each positive integer $n$ there exist odd positive integers $x_n$ and $y_n$ such that ${x_{n}}^2 +7{y_{n}}^2 =2^n$.

PEN H Problems, 72
Tags: relatively prime , induction , diophantine equation , absolute value , number theory
Find all pairs $(x, y)$ of positive rational numbers such that $x^{y}=y^{x}$.

1998 USAMTS Problems, 1
Tags: diophantine equation , number theory
Several pairs of positive integers $(m ,n )$ satisfy the condition $19m + 90 + 8n = 1998$. Of these, $(100, 1 )$ is the pair with the smallest value for $n$. Find the pair with the smallest value for $m$.

PEN H Problems, 44
Tags: calculus , diophantine equation , integration
For all $n \in \mathbb{N}$, show that the number of integral solutions $(x, y)$ of \[x^{2}+xy+y^{2}=n\] is finite and a multiple of $6$.

PEN P Problems, 15
Tags: quadratic , diophantine equation , additive number theory , algebra
Find all integers $m>1$ such that $m^3$ is a sum of $m$ squares of consecutive integers.

1999 ITAMO, 6
Tags: diophantine equation , diophantine , number theory
(a) Find all pairs $(x,k)$ of positive integers such that $3^k -1 = x^3$ . (b) Prove that if $n > 1$ is an integer, $n \ne 3$, then there are no pairs $(x,k)$ of positive integers such that $3^k -1 = x^n$.

2020 Malaysia IMONST 1, 15
Tags: diophantine equation , number theory
Find the sum of all integers $n$ that fulfill the equation \[2^n(6-n)=8n.\]

2016 Estonia Team Selection Test, 2
Tags: number theory , diophantine equation , diophantine
Let $p$ be a prime number. Find all triples $(a, b, c)$ of integers (not necessarily positive) such that $a^bb^cc^a = p$.