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: 436

2015 HMIC, 5
Tags: diophantine , cyclotomic field , roots of unity
Let $\omega = e^{2\pi i /5}$ be a primitive fifth root of unity. Prove that there do not exist integers $a, b, c, d, k$ with $k > 1$ such that \[(a + b \omega + c \omega^2 + d \omega^3)^{k}=1+\omega.\] [i]Carl Lian[/i]

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 $2p^q - q^p = 7.$

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

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 $n^2 + (n+1)^2 + (n+2)^2 = m^2$ ?

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 $3^k -1 = x^n$.

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 $a^bb^cc^a = p$.

2019 Chile National Olympiad, 3
Tags: number theory , diophantine equation , diophantine
Find all solutions $x,y,z$ in the positive integers of the equation $$3^x -5^y = z^2$$

2017 Saudi Arabia Pre-TST + Training Tests, 7
Tags: diophantine , diophantine equation , number theory
Find all pairs of integers $(x, y)$ such that $y^3 = 8x^6 + 2x^3 y -y^2$.

2018 Thailand Mathematical Olympiad, 6
Tags: diophantine , number theory , diophantine equation
Let $A$ be the set of all triples $(x, y, z)$ of positive integers satisfying $2x^2 + 3y^3 = 4z^4$ . a) Show that if $(x, y, z) \in A$ then $6$ divides all of $x, y, z$. b) Show that $A$ is an infinite set.

2004 Bosnia and Herzegovina Junior BMO TST, 1
Tags: diophantine , number theory , diophantine equation
In the set of integers solve the equation $\frac{1}{x}+\frac{1}{y}=\frac{1}{p}$, where $p$ is a prime number.

1977 Swedish Mathematical Competition, 3
Tags: diophantine , system , system of equations , number theory
Show that the only integral solution to \[\left\{ \begin{array}{l} xy + yz + zx = 3n^2 - 1\\ x + y + z = 3n \\ \end{array} \right. \] with $x \geq y \geq z$ is $x=n+1$, $y=n$, $z=n-1$.

1986 Greece Junior Math Olympiad, 1
Tags: diophantine equation , diophantine , number theory
Find all pairs of integers $(x,y)$ such that $$(x+1)(y+1)(x+y)(x^2+y^2)=16x^2y^2$$

2016 Costa Rica - Final Round, N3
Tags: number theory , diophantine equation , diophantine
Find all natural values of $n$ and $m$, such that $(n -1)2^{n - 1} + 5 = m^2 + 4m$.

2013 District Olympiad, 1
Tags: diophantine , algebra
Prove that the equation $$\frac{1}{\sqrt{x} +\sqrt{1006}}+\frac{1}{\sqrt{2012 -x} +\sqrt{1006}}=\frac{2}{\sqrt{x} +\sqrt{2012 -x}}$$ has $2013$ integer solutions.

1998 Poland - Second Round, 4
Tags: diophantine , diophantine equation , number theory
Find all pairs of integers $(x,y)$ satisfying $x^2 +3y^2 = 1998x$.

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 $a \le b \le c$.

1984 All Soviet Union Mathematical Olympiad, 379
Tags: exponential , algebra , diophantine equation , diophantine , number theory
Find integers $m$ and $n$ such that $(5 + 3 \sqrt2)^m = (3 + 5 \sqrt2)^n$.

VMEO IV 2015, 10.3
Tags: diophantine , diophantine equation , number theory
Find all triples of integers $(a, b, c)$ satisfying $a^2 + b^2 + c^2 =3(ab + bc + ca).$

2020 Switzerland - Final Round, 5
Tags: number theory , diophantine , diophantine equation , factorial
Find all the positive integers $a, b, c$ such that $$a! \cdot b! = a! + b! + c!$$

1991 Chile National Olympiad, 1
Tags: number theory , diophantine , diophantine equation
Determine all nonnegative integer solutions of the equation $2^x-2^y = 1$

2013 Costa Rica - Final Round, N1
Tags: number theory , diophantine , diophantine equation
Find all triples $(a, b, p)$ of positive integers, where $p$ is a prime number, such that $a^p - b^p = 2013$.

1987 Poland - Second Round, 5
Tags: diophantine , number theory , diophantine equation
Determine all prime numbers $ p $ and natural numbers $ x, y $ for which $ p^x-y^3 = 1 $.

2005 Estonia Team Selection Test, 3
Tags: number theory , diophantine equation , diophantine
Find all pairs $(x, y)$ of positive integers satisfying the equation $(x + y)^x = x^y$.