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

2002 Abels Math Contest (Norwegian MO), 1b
Tags: diophantine equation , diophantine , number theory
Find all integers $c$ such that the equation $(2a+b) (2b+a) =5^c$ has integer solutions.

2013 Czech And Slovak Olympiad IIIA, 1
Tags: diophantine equation , diophantine , number theory
Find all pairs of integers $a, b$ for which equality holds $\frac{a^2+1}{2b^2-3}=\frac{a-1}{2b-1}$

2013 Balkan MO Shortlist, N5
Tags: diophantine equation , prime , diophantine , number theory
Prove that there do not exist distinct prime numbers $p$ and $q$ and a positive integer $n$ satisfying the equation $p^{q-1}- q^{p-1}=4n^2$

1988 All Soviet Union Mathematical Olympiad, 471
Tags: diophantine equation , exponential , diophantine , algebra , number theory
Find all positive integers $n$ satisfying $\left(1 +\frac{1}{n}\right)^{n+1} = \left(1 + \frac{1}{1998}\right)^{1998}$.

2016 Abels Math Contest (Norwegian MO) Final, 2b
Tags: diophantine equation , diophantine , factorial , number theory
Find all non-negative integers $x, y$ and $z$ such that $x^3 + 2y^3 + 4z^3 = 9!$

2009 Philippine MO, 2
Tags: diophantine , number theory
[b](a)[/b] Find all pairs $(n,x)$ of positive integers that satisfy the equation $2^n + 1 = x^2$. [b](b)[/b] Find all pairs $(n,x)$ of positive integers that satisfy the equation $2^n = x^2 + 1$.

2016 Junior Balkan Team Selection Tests - Moldova, 6
Tags: diophantine equation , diophantine , number theory
Determine all pairs $(x, y)$ of natural numbers satisfying the equation $5^x=y^4+4y+1$.

2017 Hanoi Open Mathematics Competitions, 2
Tags: diophantine equation , number theory , diophantine
How many pairs of positive integers $(x, y)$ are there, those satisfy the identity $2^x - y^2 = 1$? (A): $1$ (B): $2$ (C): $3$ (D): $4$ (E): None of the above.

2016 Ecuador NMO (OMEC), 1
Tags: diophantine equation , diophantine
Prove that there are no positive integers $x, y$ such that: $(x + 1)^2 + (x + 2)^2 +...+ (x + 9)^2 = y^2$

1999 Singapore Team Selection Test, 1
Tags: diophantine equation , diophantine , number theory
Find all integers $m$ for which the equation $$x^3 - mx^2 + mx - (m^2 + 1) = 0$$ has an integer solution.

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

1917 Eotvos Mathematical Competition, 1
Tags: diophantine equation , system of equations , diophantine , number theory
If $a$ and $b$ are integers and if the solutions of the system of equations $$y - 2x - a = 0$$ $$y^2 - xy + x^2 - b = 0$$ are rational, prove that the solutions are integers.

2013 Balkan MO Shortlist, N6
Tags: diophantine equation , diophantine , number theory , prime
Prove that there do not exist distinct prime numbers $p$ and $q$ and a positive integer $n$ satisfying the equation $p^{q-1}- q^{p-1}=4n^3$

1962 Swedish Mathematical Competition, 3
Tags: diophantine equation , diophantine , number theory
Find all pairs $(m, n)$ of integers such that $n^2 - 3mn + m - n = 0$.

1996 Singapore MO Open, 3
Tags: diophantine equation , diophantine , number theory
Let $n$ be a positive integer. Prove that there is no positive integer solution to thxe equation $(x + 2)^n - x^n = 1 + 7^n$.

1996 Israel National Olympiad, 7
Tags: diophantine equation , diophantine , system of equations , number theory
Find all positive integers $a,b,c$ such that $$\begin{cases} a^2 = 4(b+c) \\ a^3 -2b^3 -4c^3 =\frac12 abc \end {cases}$$

2016 Abels Math Contest (Norwegian MO) Final, 2a
Tags: diophantine equation , system of equations , diophantine , number theory
Find all positive integers $a, b, c, d$ with $a \le b$ and $c \le d$ such that $\begin{cases} a + b = cd \\ c + d = ab \end{cases}$ .

1979 Czech And Slovak Olympiad IIIA, 1
Tags: diophantine equation , diophantine , number theory
Let $n$ be a given natural number. Determine the number of all orderer triples $(x, y, z)$ of non-negative integers $x, y, z$ that satisfy the equation $$x + 2y + 5z=10n.$$

1984 Swedish Mathematical Competition, 5
Tags: diophantine equation , system of equations , system , diophantine , number theory
Solve in natural numbers $a,b,c$ the system \[\left\{ \begin{array}{l}a^3 -b^3 -c^3 = 3abc \\ a^2 = 2(a+b+c)\\ \end{array} \right. \]

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

2008 Thailand Mathematical Olympiad, 7
Tags: diophantine equation , lcm , gcd , diophantine
Two positive integers $m, n$ satisfy the two equations $m^2 + n^2 = 3789$ and $gcd (m, n) + lcm (m, n) = 633$. Compute $m + n$.

2017 Switzerland - Final Round, 7
Tags: perfect square , diophantine equation , diophantine , number theory
Let $n$ be a natural number such that there are exactly$ 2017$ distinct pairs of natural numbers $(a, b)$, which the equation $$\frac{1}{a}+\frac{1}{b}=\frac{1}{n}$$ fulfilld. Show that $n$ is a perfect square . Remark: $(7, 4) \ne (4, 7)$

2019 Azerbaijan IMO TST, 3
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?

1991 Bundeswettbewerb Mathematik, 1
Tags: diophantine , diophantine equation , perfect square , number theory
Determine all solutions of the equation $4^x + 4^y + 4^z = u^2$ for integers $x,y,z$ and $u$.

2022 Austrian MO Beginners' Competition, 4
Tags: diophantine equation , number theory , diophantine
Determine all prime numbers $p, q$ and $r$ with $p + q^2 = r^4$. [i](Karl Czakler)[/i]