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

2013 Hanoi Open Mathematics Competitions, 5
Tags: number theory , diophantine equation , diophantine
The number of integer solutions $x$ of the equation below $(12x -1)(6x - 1)(4x -1)(3x - 1) = 330$ is (A): $0$, (B): $1$, (C): $2$, (D): $3$, (E): None of the above.

1993 Austrian-Polish Competition, 1
Tags: diophantine equation , diophantine , number theory
Solve in positive integers $x,y$ the equation $2^x - 3^y = 7$.

1991 Swedish Mathematical Competition, 1
Tags: diophantine equation , diophantine , number theory
Find all positive integers $m, n$ such that $\frac{1}{m} + \frac{1}{n} - \frac{1}{mn} =\frac{2}{5}$.

2007 Abels Math Contest (Norwegian MO) Final, 3
Tags: diophantine equation , diophantine , number theory
(a) Let $x$ and $y$ be two positive integers such that $\sqrt{x} +\sqrt{y}$ is an integer. Show that $\sqrt{x}$ and $\sqrt{y}$ are both integers. (b) Find all positive integers $x$ and $y$ such that $\sqrt{x} +\sqrt{y}=\sqrt{2007}$.

2000 Switzerland Team Selection Test, 7
Tags: diophantine equation , diophantine , number theory
Show that the equation $14x^2 +15y^2 = 7^{2000}$ has no integer solutions.

1977 Dutch Mathematical Olympiad, 1
Tags: number theory , diophantine , diophantine equation
Solve, for integers $x$ and $y$ : $$2x^2y = (x+2)^2(y + 1), $$ provided that $(x+2)^2(y + 1)> 1000$.

1967 Poland - Second Round, 4
Tags: diophantine , diophantine equation , number theory
Solve the equation in natural numbers $$ xy+yz+zx = xyz + 2. $$

2021 Austrian MO National Competition, 3
Tags: number theory , diophantine equation , diophantine
Find all triples $(a, b, c)$ of natural numbers $a, b$ and $c$, for which $a^{b + 20} (c-1) = c^{b + 21} - 1$ is satisfied. (Walther Janous)

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

1998 German National Olympiad, 3
Tags: diophantine , diophantine equation , number theory
For each nonnegative integer $k$ find all nonnegative integers $x,y,z$ such that $x^2 +y^2 +z^2 = 8^k$

2015 Swedish Mathematical Competition, 2
Tags: diophantine , diophantine equation , number theory
Determine all integer solutions to the equation $x^3 + y^3 + 2015 = 0$.

2019 Thailand TST, 2
Tags: number theory , diophantine
Four positive integers $x,y,z$ and $t$ satisfy the relations \[ xy - zt = x + y = z + t. \] Is it possible that both $xy$ and $zt$ are perfect squares?

2002 Dutch Mathematical Olympiad, 2
Tags: diophantine equation , diophantine , number theory
Determine all triplets $(x, y, z)$ of positive integers with $x \le y \le z$ that satisfy $\left(1+\frac1x \right)\left(1+\frac1y \right)\left(1+\frac1z \right) = 3$

2014 Hanoi Open Mathematics Competitions, 11
Tags: number theory , diophantine equation , diophantine
Find all pairs of integers $(x,y)$ satisfying the following equality $8x^2y^2 + x^2 + y^2 = 10xy$

2015 Saudi Arabia Pre-TST, 2.3
Tags: diophantine , diophantine equation , number theory
Find all integer solutions of the equation $14^x - 3^y = 2015$. (Malik Talbi)

1999 Austrian-Polish Competition, 7
Tags: diophantine , diophantine equation , number theory
Find all pairs $(x,y)$ of positive integers such that $x^{x+y} =y^{y-x}$.

2017 Hanoi Open Mathematics Competitions, 6
Tags: number theory , diophantine equation , prime , diophantine
Find all triples of positive integers $(m,p,q)$ such that $2^mp^2 + 27 = q^3$ and $p$ is a prime.

2017 Singapore Junior Math Olympiad, 2
Tags: diophantine , diophantine equation , number theory , factorial
Let $n$ be a positive integer and $a_1,a_2,...,a_{2n}$ be $2n$ distinct integers. Given that the equation $|x-a_1| |x-a_2| ... |x-a_{2n}| =(n!)^2$ has an integer solution $x = m$, find $m$ in terms of $a_1,a_2,...,a_{2n}$

1990 Austrian-Polish Competition, 4
Tags: system of equations , number theory , diophantine equation , diophantine
Find all solutions in positive integers to: $$\begin{cases} x_1^4 + 14x_1x_2 + 1 = y_1^4 \\ x_2^4 + 14x_2x_3 + 1 = y_2^4 \\ ... \\ x_n^4 + 14x_nx_1 + 1 = y_n^4 \end{cases}$$

1987 Dutch Mathematical Olympiad, 1
Tags: number theory , diophantine , diophantine equation
Solve into $N$: $$a^2 = 2^b +c^4$$

2008 Hanoi Open Mathematics Competitions, 2
Tags: diophantine equation , diophantine , number theory
Find all pairs $(m, n)$ of positive integers such that $m^2 + 2n^2 = 3(m + 2n)$

2021 Malaysia IMONST 1, 12
Tags: number theory , diophantine equation , diophantine
Determine the number of positive integer solutions $(x,y, z)$ to the equation $xyz = 2(x + y + z)$.

Russian TST 2019, P3
Tags: number theory , diophantine
Four positive integers $x,y,z$ and $t$ satisfy the relations \[ xy - zt = x + y = z + t. \] Is it possible that both $xy$ and $zt$ are perfect squares?

VMEO III 2006 Shortlist, N5
Tags: number theory , diophantine equation , diophantine
Find all triples of integers $(x, y, z)$ such that $x^4 + 5y^4 = z^4$.

2022 Dutch IMO TST, 1
Tags: number theory , diophantine equation , diophantine
Find all quadruples $(a, b, c, d)$ of non-negative integers such that $ab =2(1 + cd)$ and there exists a non-degenerate triangle with sides of length $a - c$, $b - d$, and $c + d$.