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

2023 Puerto Rico Team Selection Test, 1
Tags: diophantine , diophantine equation , number theory
Determine all triples $(a, b, c)$ of positive integers such that $$a! +b! = 2^{c!} .$$

2006 IMO Shortlist, 1
Tags: diophantine equation , number theory
Determine all pairs $(x, y)$ of integers such that \[1+2^{x}+2^{2x+1}= y^{2}.\]

2017 Kyiv Mathematical Festival, 5
Tags: number theory , diophantine equation
Find all the pairs of integers $(x,y)$ for which $(x^2+y)(y^2+x)=(x+1)(y+1).$

2014 Ukraine Team Selection Test, 9
Tags: number theory , diophantine , prime number , diophantine equation , parameterization , system of equations
Let $m, n$ be odd prime numbers. Find all pairs of integers numbers $a, b$ for which the system of equations: $x^m+y^m+z^m=a$, $x^n+y^n+z^n=b$ has many solutions in integers $x, y, z$.

2023 Olympic Revenge, 2
Tags: diophantine equation , number theory
Find all triples ($a$,$b$,$n$) of positive integers such that $$a^3=b^2+2^n$$

2022 Czech-Polish-Slovak Junior Match, 2
Tags: diophantine , diophantine equation , number theory
Solve the following system of equations in integer numbers: $$\begin{cases} x^2 = yz + 1 \\ y^2 = zx + 1 \\ z^2 = xy + 1 \end{cases}$$

2013 Junior Balkan Team Selection Tests - Moldova, 6
Tags: diophantine , diophantine equation , number theory
Determine all triplets of real numbers $(x, y, z)$ that satisfy the equation $4xyz = x^4 + y^4 + z^4 + 1$.

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

2016 Korea Winter Program Practice Test, 1
Tags: diophantine equation , number theory
Solve: $a, b, m, n\in \mathbb{N}$ $a^2+b^2=m^2-n^2, ab=2mn$

2013 Danube Mathematical Competition, 3
Tags: exponential , number theory , diophantine , diophantine equation , exponential equation
Determine the natural numbers $m,n$ such as $85^m-n^4=4$

2007 Indonesia TST, 3
Tags: inequalities , diophantine equation , floor function , number theory
For each real number $ x$< let $ \lfloor x \rfloor$ be the integer satisfying $ \lfloor x \rfloor \le x < \lfloor x \rfloor \plus{}1$ and let $ \{x\}\equal{}x\minus{}\lfloor x \rfloor$. Let $ c$ be a real number such that \[ \{n\sqrt{3}\}>\dfrac{c}{n\sqrt{3}}\] for all positive integers $ n$. Prove that $ c \le 1$.

2015 USAMO, 1
Tags: diophantine equation , number theory , algebra
Solve in integers the equation \[ x^2+xy+y^2 = \left(\frac{x+y}{3}+1\right)^3. \]

1973 Swedish Mathematical Competition, 4
Tags: diophantine , number theory , diophantine equation
$p$ is a prime. Find all relatively prime positive integers $m$, $n$ such that \[ \frac{m}{n}+\frac{1}{p^2}=\frac{m+p}{n+p} \]

2019 Romania National Olympiad, 4
Tags: diophantine , diophantine equation , number theory
Find the natural numbers $x, y, z$ that verify the equation: $$2^x + 3 \cdot 11^y =7^z$$

2009 IMAR Test, 1
Tags: number theory , diophantine equation , algebra , system of equations
Given $a$ and $b$ distinct positive integers, show that the system of equations $x y +zw = a$ $xz + yw = b$ has only finitely many solutions in integers $x, y, z,w$.

2006 Singapore Junior Math Olympiad, 1
Tags: number theory , diophantine , diophantine equation
Find all integers $x,y$ that satisfy the equation $x+y=x^2-xy+y^2$

2017 Danube Mathematical Olympiad, 4
Tags: number theory , diophantine equation
Determine all triples of positive integers $(x,y,z)$ such that $x^4+y^4 =2z^2$ and $x$ and $y$ are relatively prime.

1967 Swedish Mathematical Competition, 3
Tags: diophantine , diophantine equation , number theory
Show that there are only finitely many triples $(a, b, c)$ of positive integers such that $\frac{1}{a} + \frac{1}{b} + \frac{1}{c} = \frac{1}{1000}$.

2008 Hanoi Open Mathematics Competitions, 4
Tags: diophantine , diophantine equation , number theory
Find all pairs $(m,n)$ of positive integers such that $m^2 + n^2 = 3(m + n)$.

2019 Durer Math Competition Finals, 15
Tags: diophantine , diophantine equation , sum , number theory
The positive integer $m$ and non-negative integers $x_0, x_1,..., x_{1001}$ satisfy the following equation: $$m^{x_0} =\sum_{i=1}^{1001}m^{x_i}.$$ How many possibilities are there for the value of $m$?

2017 Greece Junior Math Olympiad, 3
Tags: diophantine , diophantine equation , number theory
Find all triplets $(a,b,p)$ where $a,b$ are positive integers and $p$ is a prime number satisfying: $\frac{1}{p}=\frac{1}{a^2}+\frac{1}{b^2}$

2014 CHKMO, 3
Tags: diophantine equation , number theory
Find all pairs $(a,b)$ of integers $a$ and $b$ satisfying \[(b^2+11(a-b))^2=a^3 b\]

2001 Croatia Team Selection Test, 3
Tags: diophantine equation , number theory , diophantine
Find all solutions of the equation $(a^a)^5 = b^b$ in positive integers.

1998 Swedish Mathematical Competition, 1
Tags: diophantine , diophantine equation , number theory
Find all positive integers $a, b, c$, such that $(8a-5b)^2 + (3b-2c)^2 + (3c-7a)^2 = 2$.

PEN H Problems, 76
Tags: diophantine equation
Find all pairs $(m,n)$ of integers that satisfy the equation \[(m-n)^{2}=\frac{4mn}{m+n-1}.\]