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

2022 Austrian MO National Competition, 4
Tags: prime number , number theory , diophantine equation , diophantine
Find all triples $(p, q, r)$ of prime numbers for which $4q - 1$ is a prime number and $$\frac{p + q}{p + r} = r - p$$ holds. [i](Walther Janous)[/i]

1993 Romania Team Selection Test, 4
Tags: number theory , diophantine
Prove that the equation $ (x\plus{}y)^n\equal{}x^m\plus{}y^m$ has a unique solution in integers with $ x>y>0$ and $ m,n>1$.

2020 Regional Competition For Advanced Students, 4
Tags: number theory , diophantine equation , diophantine
Find all quadruples $(p, q, r, n)$ of prime numbers $p, q, r$ and positive integer numbers $n$, such that $$p^2 = q^2 + r^n$$ (Walther Janous)

1994 Tournament Of Towns, (413) 1
Tags: number theory , diophantine , diophantine equation
Does there exist an infinite set of triples of integers $x, y, z$ (not necessarily positive) such that $$x^2 + y^2 + z^2 = x^3 + y^3+z^3?$$ (NB Vassiliev)

2017 Istmo Centroamericano MO, 3
Tags: number theory , diophantine equation , diophantine
Find all ordered pairs of integers $(x, y)$ with $y \ge 0$ such that $x^2 + 2xy + y! = 131$.

2021 Grand Duchy of Lithuania, 4
Tags: number theory , diophantine equation , diophantine
A triplet of positive integers $(x, y, z)$ satisfying $x, y, z > 1$ and $x^3 - yz^3 = 2021$ is called [i]primary [/i] if at least two of the integers $x, y, z$ are prime numbers. a) Find at least one primary triplet. b) Show that there are infinitely many primary triplets.

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

1999 ITAMO, 6
Tags: diophantine equation , number theory , diophantine
(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 $3^k -1 = x^n$.

2014 Hanoi Open Mathematics Competitions, 10
Tags: number theory , diophantine equation , diophantine
Find all pairs of integers $(x, y)$ satisfying the condition $12x^2 + 6xy + 3y^2 = 28(x + y)$.

2011 Swedish Mathematical Competition, 1
Tags: number theory , diophantine equation , diophantine
Determine all positive integers $k$, $\ell$, $m$ and $n$, such that $$\frac{1}{k!}+\frac{1}{\ell!}+\frac{1}{m!} =\frac{1}{n!} $$

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

2014 Swedish Mathematical Competition, 6
Tags: number theory , diophantine , diophantine equation
Determine all odd primes $p$ and $q$ such that the equation $x^p + y^q = pq$ at least one solution $(x, y)$ where $x$ and $y$ are positive integers.

2015 Hanoi Open Mathematics Competitions, 14
Tags: diophantine equation , number theory , diophantine
Determine all pairs of integers $(x, y)$ such that $2xy^2 + x + y + 1 = x^2 + 2y^2 + xy$.

1997 Estonia National Olympiad, 1
Tags: number theory , diophantine , diophantine equation
Prove that for every integer $n\ge 3$ there are such positives integers $x$ and $y$ such that $2^n = 7x^2 + y^2$

1997 Tournament Of Towns, (558) 3
Tags: number theory , diophantine , diophantine equation
Prove that the equation $$xy(x -y) + yz(y-z) + zx(z-x) = 6$$ has infinitely many solutions in integers $x, y$ and $z$. (N Vassiliev)

2022 Dutch IMO TST, 1
Tags: number theory , diophantine equation , diophantine
Determine all positive integers $n \ge 2$ which have a positive divisor $m | n$ satisfying $$n = d^3 + m^3.$$ where $d$ is the smallest divisor of $n$ which is greater than $1$.

1996 Greece Junior Math Olympiad, 4b
Tags: diophantine equation , diophantine , prime , number theory
Determine whether exist a prime number $p$ and natural number $n$ such that $n^2 + n + p = 1996$.

2021 Federal Competition For Advanced Students, P2, 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)

1980 Polish MO Finals, 2
Tags: number theory , diophantine , diophantine equation
Prove that for every $n$ there exists a solution of the equation $$a^2 +b^2 +c^2 = 3abc$$ in natural numbers $a,b,c$ greater than $n$.

2004 Estonia National Olympiad, 1
Tags: diophantine equation , diophantine , number theory
Find all pairs of integers $(a, b)$ such that $a^2 + ab + b^2 = 1$

1994 Abels Math Contest (Norwegian MO), 2b
Tags: diophantine equation , diophantine , number theory
Find all integers $x,y,z$ such that $x^3 +5y^3 = 9z^3$.

2002 Argentina National Olympiad, 2
Tags: number theory , diophantine equation , diophantine
Determine the smallest positive integer $k$ so that the equation $$2002x+273y=200201+k$$ has integer solutions, and for that value of $k$, find the number of solutions $\left (x,y\right )$ with $x$, $y$ positive integers that have the equation.

1993 All-Russian Olympiad Regional Round, 9.5
Tags: number theory , diophantine
Show that the equation $x^3 +y^3 = 4(x^2y+xy^2 +1)$ has no integer solutions.

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

2010 Saudi Arabia Pre-TST, 2.1
Tags: diophantine , system of equations , system , number theory
Find all triples $(x,y,z)$ of positive integers such that $$\begin{cases} x + y +z = 2010 \\x^2 + y^2 + z^2 - xy - yz - zx =3 \end{cases}$$