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

2014 Thailand TSTST, 1
Tags: diophantine equation , number theory
Find all triples of positive integers $(a, b, c)$ such that $$(2^a-1)(3^b-1)=c!.$$

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 $m + n$.

2024 Baltic Way, 20
Tags: diophantine equation , discriminant , number theory
Positive integers $a$, $b$ and $c$ satisfy the system of equations \begin{align*} (ab-1)^2&=c(a^2+b^2)+ab+1,\\ a^2+b^2&=c^2+ab. \end{align*} a) Prove that $c+1$ is a perfect square. b) Find all such triples $(a,b,c)$.

2019 Vietnam TST, P4
Tags: number theory , diophantine equation
Find all triplets of positive integers $(x, y, z)$ such that $2^x+1=7^y+2^z$.

2023 Dutch BxMO TST, 5
Tags: modular arithmetic , diophantine equation , number theory
Find all pairs of prime numbers $(p,q)$ for which \[2^p = 2^{q-2} + q!.\]

1990 French Mathematical Olympiad, Problem 3
Tags: number theory , diophantine equation
(a) Find all triples of integers $(a,b,c)$ for which $\frac14=\frac1{a^2}+\frac1{b^2}+\frac1{c^2}$. (b) Determine all positive integers $n$ for which there exist positive integers $x_1,x_2,\ldots,x_n$ such that $1=\frac1{x_1^2}+\frac1{x_2^2}+\ldots+\frac1{x_n^2}$.

2000 Swedish Mathematical Competition, 3
Tags: number theory , diophantine equation , diophantine
Are there any integral solutions to $n^2 + (n+1)^2 + (n+2)^2 = m^2$ ?

PEN H Problems, 11
Tags: diophantine equation
Find all $(x,y,n) \in {\mathbb{N}}^3$ such that $\gcd(x, n+1)=1$ and $x^{n}+1=y^{n+1}$.

PEN H Problems, 73
Tags: greatest common divisor , diophantine equation
Find all pairs $(a,b)$ of positive integers that satisfy the equation \[a^{b^{2}}= b^{a}.\]

2013 Estonia Team Selection Test, 1
Tags: number theory , system of equations , diophantine , diophantine equation
Find all prime numbers $p$ for which one can find a positive integer $m$ and nonnegative integers $a_0,a_1,...,a_m$ less than $p$ such that $$\begin{cases} a_0+a_1p+...+a_{m-1}p^{m-1}+a_{m}p^{m} = 2013 \\ a_0+a_1+...+a_{m-1}+a_{m} = 11\end{cases}$$

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

2010 ELMO Shortlist, 3
Tags: quadratic , diophantine equation , number theory
Prove that there are infinitely many quadruples of integers $(a,b,c,d)$ such that \begin{align*} a^2 + b^2 + 3 &= 4ab\\ c^2 + d^2 + 3 &= 4cd\\ 4c^3 - 3c &= a \end{align*} [i]Travis Hance.[/i]

2023 Singapore Junior Math Olympiad, 5
Tags: number theory , diophantine equation
Find all positive integers $k$ such that there exists positive integers $a, b$ such that \[a^2+4=(k^2-4)b^2.\]

2018 Korea Junior Math Olympiad, 7
Tags: number theory , diophantine equation
Find all integer pair $(m,n)$ such that $7^m=5^n+24$.

2001 Hungary-Israel Binational, 1
Tags: diophantine equation , number theory
Find positive integers $x, y, z$ such that $x > z > 1999 \cdot 2000 \cdot 2001 > y$ and $2000x^{2}+y^{2}= 2001z^{2}.$

2022 VTRMC, 3
Tags: number theory , diophantine equation
Find all positive integers $a, b, c, d,$ and $n$ satisfying $n^a + n^b + n^c = n^d$ and prove that these are the only such solutions.

2006 IMO, 4
Tags: number theory , diophantine equation
Determine all pairs $(x, y)$ of integers such that \[1+2^{x}+2^{2x+1}= y^{2}.\]

2020 Malaysia IMONST 1, 15
Tags: diophantine equation , number theory
Find the sum of all integers $n$ that fulfill the equation \[2^n(6-n)=8n.\]

PEN H Problems, 67
Tags: diophantine equation
Is there a positive integer $m$ such that the equation \[\frac{1}{a}+\frac{1}{b}+\frac{1}{c}+\frac{1}{abc}= \frac{m}{a+b+c}\] has infinitely many solutions in positive integers $a, b, c \;$?

2007 Swedish Mathematical Competition, 1
Tags: number theory , diophantine equation , system of equations , diophantine
Solve the following system \[ \left\{ \begin{array}{l} xyzu-x^3=9 \\ x+yz=\dfrac{3}{2}u \\ \end{array} \right. \] in positive integers $x$, $y$, $z$ and $u$.

1992 Bulgaria National Olympiad, Problem 4
Tags: number theory , diophantine equation
Let $p$ be a prime number in the form $p=4k+3$. Prove that if the numbers $x_0,y_0,z_0,t_0$ are solutions of the equation $x^{2p}+y^{2p}+z^{2p}=t^{2p}$, then at least one of them is divisible by $p$. [i](Plamen Koshlukov)[/i]

VI Soros Olympiad 1999 - 2000 (Russia), 9.4
Tags: number theory , diophantine equation , diophantine
Are there integers $k$ and $m$ for which $$\frac{(k-3)(k-2)(k-1)k+1}{(k+1)(k+2)(k+3)(k+4)+1}=m(m+1)+(m+1)(m+2)+(m+2)m \,\, ?$$

2024 Bundeswettbewerb Mathematik, 1
Tags: diophantine equation , number theory
Determine all pairs $(x,y)$ of integers satisfying \[(x+2)^4-x^4=y^3.\]

2005 Estonia Team Selection Test, 3
Tags: diophantine equation , diophantine , number theory
Find all pairs $(x, y)$ of positive integers satisfying the equation $(x + y)^x = x^y$.

2014 Estonia Team Selection Test, 6
Tags: diophantine , diophantine equation , number theory
Find all natural numbers $n$ such that the equation $x^2 + y^2 + z^2 = nxyz$ has solutions in positive integers