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

PEN H Problems, 23
Tags: inequalities , diophantine equation
Find all $(x,y,z) \in {\mathbb{Z}}^3$ such that $x^{3}+y^{3}+z^{3}=x+y+z=3$.

2014 NIMO Problems, 15
Tags: diophantine equation
Let $A = (0,0)$, $B=(-1,-1)$, $C=(x,y)$, and $D=(x+1,y)$, where $x > y$ are positive integers. Suppose points $A$, $B$, $C$, $D$ lie on a circle with radius $r$. Denote by $r_1$ and $r_2$ the smallest and second smallest possible values of $r$. Compute $r_1^2 + r_2^2$. [i]Proposed by Lewis Chen[/i]

2000 Switzerland Team Selection Test, 7
Tags: diophantine equation , diophantine , number theory
Show that the equation $14x^2 +15y^2 = 7^{2000}$ has no integer solutions.

2019 Azerbaijan Junior NMO, 5
Tags: diophantine equation , number theory
Find all $x;y\in\mathbb{Z}$ satisfying the following condition: $$x^3=y^4+9x^2$$

2007 India IMO Training Camp, 2
Tags: modular arithmetic , diophantine equation , number theory
Find all integer solutions $(x,y)$ of the equation $y^2=x^3-p^2x,$ where $p$ is a prime such that $p\equiv 3 \mod 4.$

2022 USAMO, 4
Tags: number theory , diophantine equation , prime number
Find all pairs of primes $(p, q)$ for which $p-q$ and $pq-q$ are both perfect squares.

2009 QEDMO 6th, 1
Tags: diophantine , diophantine equation , number theory
Solve $y^5 - x^2 = 4$ in integers numbers $x,y$.

2020 Korea National Olympiad, 4
Tags: diophantine equation , number theory
Find a pair of coprime positive integers $(m,n)$ other than $(41,12)$ such that $m^2-5n^2$ and $m^2+5n^2$ are both perfect squares.

2014 Hanoi Open Mathematics Competitions, 11
Tags: number theory , diophantine equation , diophantine
Find all pairs of integers $(x,y)$ satisfying the following equality $8x^2y^2 + x^2 + y^2 = 10xy$

2003 Abels Math Contest (Norwegian MO), 2a
Tags: diophantine equation , diophantine , number theory
Find all pairs $(x, y)$ of integers numbers such that $y^3+5=x(y^2+2)$

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.

PEN H Problems, 59
Tags: modular arithmetic , diophantine equation
Solve the equation $28^x =19^y +87^z$, where $x, y, z$ are integers.

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 $x, y,z,t$.

1995 India National Olympiad, 2
Tags: quadratic , number theory , relatively prime , diophantine equation , algebra
Show that there are infintely many pairs $(a,b)$ of relatively prime integers (not necessarily positive) such that both the equations \begin{eqnarray*} x^2 +ax +b &=& 0 \\ x^2 + 2ax + b &=& 0 \\ \end{eqnarray*} have integer roots.

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

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

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

1965 Swedish Mathematical Competition, 2
Tags: number theory , diophantine equation , diophantine
Find all positive integers m, n such that $m^3 - n^3 = 999$.

PEN H Problems, 13
Tags: modular arithmetic , geometry , 3d geometry , diophantine equation
Find all pairs $(x,y)$ of positive integers that satisfy the equation \[y^{2}=x^{3}+16.\]

2022 Dutch BxMO TST, 3
Tags: diophantine equation , diophantine , number theory
Find all pairs $(p, q)$ of prime numbers such that $$p(p^2 -p - 1) = q(2q + 3).$$

2012 AMC 10, 22
Tags: modular arithmetic , number theory , diophantine equation , algebra , system of equations
The sum of the first $m$ positive odd integers is $212$ more than the sum of the first $n$ positive even integers. What is the sum of all possible values of $n$? $ \textbf{(A)}\ 255 \qquad\textbf{(B)}\ 256 \qquad\textbf{(C)}\ 257 \qquad\textbf{(D)}\ 258 \qquad\textbf{(E)}\ 259 $

2014 Stars Of Mathematics, 1
Tags: number theory , diophantine equation
Prove there exist infinitely many pairs $(x,y)$ of integers $1<x<y$, such that $x^3+y \mid x+y^3$. ([i]Dan Schwarz[/i])

1998 Estonia National Olympiad, 4
Tags: factorial , diophantine , diophantine equation , number theory
Find all integers $n > 2$ for which $(2n)! = (n-2)!n!(n+2)!$ .

2011 Croatia Team Selection Test, 4
Tags: number theory , relatively prime , diophantine equation , equation
Find all pairs of integers $x,y$ for which \[x^3+x^2+x=y^2+y.\]

Russian TST 2016, P1
Tags: number theory , diophantine equation
Find all $ x, y, z\in\mathbb{Z}^+ $ such that \[ (x-y)(y-z)(z-x)=x+y+z \]