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

2012 Dutch Mathematical Olympiad, 3
Tags: number theory , diophantine equation , diophantine
Determine all pairs $(p,m)$ consisting of a prime number $p$ and a positive integer $m$, for which $p^3 + m(p + 2) = m^2 + p + 1$ holds.

PEN H Problems, 43
Tags: diophantine equation
Find all solutions in integers of $x^{3}+2y^{3}=4z^{3}$.

2017 Thailand Mathematical Olympiad, 7
Tags: diophantine equation , diophantine , number theory
Show that no pairs of integers $(m, n)$ satisfy $2560m^2 + 5m + 6 = n^5$. .

2009 IMAR Test, 1
Tags: diophantine equation , algebra , number theory , 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 $x, y, z,w$.

2015 BMT Spring, 9
Tags: number theory , diophantine equation
There exists a unique pair of positive integers $k,n$ such that $k$ is divisible by $6$, and $\sum_{i=1}^ki^2=n^2$. Find $(k,n)$.

1999 Abels Math Contest (Norwegian MO), 2a
Tags: diophantine equation , diophantine , number theory
Find all integers $m$ and $n$ such that $2m^2 +n^2 = 2mn+3n$

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

2019 Austrian Junior Regional Competition, 1
Tags: number theory , diophantine equation , diophantine
Let $x$ and $y$ be integers with $x + y \ne 0$. Find all pairs $(x, y)$ such that $$\frac{x^2 + y^2}{x + y}= 10.$$ (Walther Janous)

1981 IMO, 3
Tags: algebra , polynomial , vieta , number theory , diophantine equation
Determine the maximum value of $m^2+n^2$, where $m$ and $n$ are integers in the range $1,2,\ldots,1981$ satisfying $(n^2-mn-m^2)^2=1$.

2008 Hanoi Open Mathematics Competitions, 3
Tags: number theory , diophantine , diophantine equation
Show that the equation $x^2 + 8z = 3 + 2y^2$ has no solutions of positive integers $x, y$ and $z$.

1966 Polish MO Finals, 1
Tags: diophantine , diophantine equation , number theory
Solve in integers the equation $$x^4 +4y^4 = 2(z^4 +4u^4)$$

2008 India National Olympiad, 2
Tags: quadratic , modular arithmetic , number theory , equation , diophantine equation
Find all triples $ \left(p,x,y\right)$ such that $ p^x\equal{}y^4\plus{}4$, where $ p$ is a prime and $ x$ and $ y$ are natural numbers.

2014 China Team Selection Test, 3
Tags: number theory , diophantine equation
Show that there are no 2-tuples $ (x,y)$ of positive integers satisfying the equation $ (x+1) (x+2)\cdots (x+2014)= (y+1) (y+2)\cdots (y+4028).$

2008 Dutch Mathematical Olympiad, 2
Tags: number theory , diophantine equation , diophantine
Find all positive integers $(m, n)$ such that $3 \cdot 2^n + 1 = m^2$.

1967 IMO Longlists, 38
Tags: modular arithmetic , number theory , perfect cube , diophantine equation
Does there exist an integer such that its cube is equal to $3n^2 + 3n + 7,$ where $n$ is an integer.

1926 Eotvos Mathematical Competition, 1
Tags: diophantine , diophantine equation , system of equations , system , number theory
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 $x, y,z,t$.

2017 India IMO Training Camp, 2
Tags: number theory , diophantine equation
Find all positive integers $p,q,r,s>1$ such that $$p!+q!+r!=2^s.$$

2002 Singapore Senior Math Olympiad, 3
Tags: diophantine , radical , number theory , diophantine equation
Prove that for natural numbers $p$ and $q$, there exists a natural number $x$ such that $$(\sqrt{p}+\sqrt{p-1})^q=\sqrt{x}+\sqrt{x-1}$$ (As an example, if $p = 3, q = 2$, then $x$ can be taken to be $25$.)

I Soros Olympiad 1994-95 (Rus + Ukr), 9.5
Tags: number theory , diophantine equation , diophantine
Find the triplets of natural numbers $(p,q,r)$ that satisfy the equality $$\frac{1}{p}+\frac{q}{q^r -1}=1.$$

2017 Simon Marais Mathematical Competition, B2
Tags: modular arithmetic , diophantine equation , number theory
Find all prime numbers $p,q$, for which $p^{q+1}+q^{p+1}$ is a perfect square. [i]Proposed by P. Boyvalenkov[/i]

1989 IMO Longlists, 8
Tags: algebra , polynomial , root , equation , diophantine equation
Find the roots $ r_i \in \mathbb{R}$ of the polynomial \[ p(x) \equal{} x^n \plus{} n \cdot x^{n\minus{}1} \plus{} a_2 \cdot x^{n\minus{}2} \plus{} \ldots \plus{} a_n\] satisfying \[ \sum^{16}_{k\equal{}1} r^{16}_k \equal{} n.\]

2016 Latvia Baltic Way TST, 17
Tags: number theory , diophantine equation , diophantine , prime
Can you find five prime numbers $p, q, r, s, t$ such that $p^3+q^3+r^3+s^3 =t^3$?

PEN H Problems, 88
Tags: diophantine equation , induction , number theory
(Leo Moser) Show that the Diophantine equation \[\frac{1}{x_{1}}+\frac{1}{x_{2}}+\cdots+\frac{1}{x_{n}}+\frac{1}{x_{1}x_{2}\cdots x_{n}}= 1\] has at least one solution for every positive integers $n$.

2009 Bulgaria National Olympiad, 1
Tags: number theory , diophantine equation , exponential equation , coprime
The natural numbers $a$ and $b$ satis fy the inequalities $a > b > 1$ . It is also known that the equation $\frac{a^x - 1}{a - 1}=\frac{b^y - 1}{b - 1}$ has at least two solutions in natural numbers, when $x > 1$ and $y > 1$. Prove that the numbers $a$ and $b$ are coprime (their greatest common divisor is $1$).

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