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

1997 Denmark MO - Mohr Contest, 4
Tags: number theory , diophantine , diophantine equation
Find all pairs $x,y$ of natural numbers that satisfy the equation $$x^2-xy+2x-3y=1997$$

1926 Eotvos Mathematical Competition, 1
Tags: system of equations , number theory , diophantine , diophantine equation , system
Prove that, if $a$ and $b$ are given integers, the system of equatìons $$x + y + 2z + 2t = a$$ $$2x - 2y + z- t = b$$ has a solution in integers $x, y,z,t$.

1993 Tournament Of Towns, (389) 1
Tags: diophantine , diophantine equation , number theory
Consider the set of solutions of the equation $$x^2+y^3=z^2.$$ in positive integers. Is it finite or infinite? (Folklore)

1968 All Soviet Union Mathematical Olympiad, 107
Tags: number theory , diophantine
Prove that the equation $x^2 + x + 1 = py$ has solution $(x,y)$ for the infinite number of simple $p$.

2013 Switzerland - Final Round, 9
Tags: diophantine , diophantine equation , number theory
Find all quadruples $(p, q, m, n)$ of natural numbers such that $p$ and $q$ are prime and the the following equation is fulfilled: $$p^m - q^3 = n^3$$

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)

1948 Moscow Mathematical Olympiad, 148
Tags: number theory , algebra , diophantine , exponential
a) Find all positive integer solutions of the equation $x^y = y^x$ ($x \ne y$). b) Find all positive rational solutions of the equation $x^y = y^x$ ($x \ne y$).

2010 Saudi Arabia BMO TST, 1
Tags: diophantine , number theory , diophantine equation
Find all triples $(x,y,z)$ of positive integers such that $3^x + 4^y = 5^z$.

2017 QEDMO 15th, 1
Tags: number theory , diophantine , diophantine equation
Find all integers $x, y, z$ satisfy the $x^4-10y^4 + 3z^6 = 21$.

Mathley 2014-15, 2
Tags: diophantine , number theory , diophantine equation
Let $n$ be a positive integer and $p$ a prime number $p > n + 1$. Prove that the following equation does not have integer solution $$1 + \frac{x}{n + 1} + \frac{x^2}{2n + 1} + ...+ \frac{x^p}{pn + 1} = 0$$ Luu Ba Thang, Department of Mathematics, College of Education

2016 Costa Rica - Final Round, N3
Tags: number theory , diophantine equation , diophantine
Find all nonnegative integers $a$ and $b$ that satisfy the equation $$3 \cdot 2^a + 1 = b^2.$$

1995 Israel Mathematical Olympiad, 4
Tags: diophantine , diophantine equation , number theory
Find all integers $m$ and $n$ satisfying $m^3 -n^3 - 9mn = 27$.

2021 Final Mathematical Cup, 1
Tags: diophantine , number theory , diophantine equation
Find all integer $n$ such that the equation $2x^2 + 5xy + 2y^2 = n$ has integer solution for $x$ and $y$.

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 $c + d$.

1983 Swedish Mathematical Competition, 3
Tags: diophantine , number theory , even , system of equations , system , diophantine equation
The systems of equations \[\left\{ \begin{array}{l} 2x_1 - x_2 = 1 \\ -x_1 + 2x_2 - x_3 = 1 \\ -x_2 + 2x_3 - x_4 = 1 \\ -x_3 + 3x_4 - x_5 =1 \\ \cdots\cdots\cdots\cdots\\ -x_{n-2} + 2x_{n-1} - x_n = 1 \\ -x_{n-1} + 2x_n = 1 \\ \end{array} \right. \] has a solution in positive integers $x_i$. Show that $n$ must be even.

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

2022 Argentina National Olympiad, 5
Tags: diophantine , number theory , diophantine equation
Find all pairs of positive integers $x,y$ such that $$x^3+y^3=4(x^2y+xy^2-5).$$

2005 All-Russian Olympiad Regional Round, 8.7
Tags: number theory , diophantine , system of equations
Find all pairs $(x, y)$ of natural numbers such that $$x + y = a^n, x^2 + y^2 = a^m$$ for some natural $a, n, m$.

2011 NZMOC Camp Selection Problems, 4
Tags: number theory , diophantine , diophantine equation
Find all pairs of positive integers $m$ and $n$ such that $$(m + 1)! + (n + 1)! = m^2n.$$

2015 China Northern MO, 1
Tags: number theory , diophantine equation , diophantine
Find all integer solutions to the equation $$\frac{xyz}{w}+\frac{yzw}{x}+\frac{zwx}{y}+\frac{wxy}{z}=4$$

2019 Hanoi Open Mathematics Competitions, 11
Tags: number theory , diophantine , diophantine equation
Find all integers $x$ and $y$ satisfying the following equation $x^2 - 2xy + 5y^2 + 2x - 6y - 3 = 0$.

1999 Swedish Mathematical Competition, 3
Tags: number theory , diophantine equation , diophantine
Find non-negative integers $a, b, c, d$ such that $5^a + 6^b + 7^c + 11^d = 1999$.

2011 NZMOC Camp Selection Problems, 1
Tags: number theory , diophantine , diophantine equation
Find all pairs of positive integers $m$ and $n$ such that $$m! + n! = m^n.$$ .

1998 Singapore MO Open, 3
Tags: number theory , diophantine , diophantine equation
Do there exist integers $x$ and $y$ such that $19^{19} = x^3 +y^4$ ? Justify your answer.

2014 Contests, 1
Tags: number theory , diophantine equation , diophantine
Determine all triples $(a,b,c)$, where $a, b$, and $c$ are positive integers that satisfy $a \le b \le c$ and $abc = 2(a + b + c)$.