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

2008 Regional Olympiad of Mexico Center Zone, 1

Find all pairs of integers $ a, b $ that satisfy $a ^2-3a = b ^3-2$.

2008 India National Olympiad, 2

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, 60

Show that the equation $x^7 + y^7 = {1998}^z$ has no solution in positive integers.

2010 Contests, 3

Determine all $(m,n) \in \mathbb{Z}^+ \times \mathbb{Z}^+$ which satisfy $3^m-7^n=2.$

2013 Balkan MO Shortlist, N3

Determine all quadruplets ($x, y, z, t$) of positive integers, such that $12^x + 13^y - 14^z = 2013^t$.

2015 FYROM JBMO Team Selection Test, 1

Solve the equation $x^2+y^4+1=6^z$ in the set of integers.

PEN H Problems, 45

Show that there cannot be four squares in arithmetical progression.

1986 IMO Shortlist, 4

Provided the equation $xyz = p^n(x + y + z)$ where $p \geq 3$ is a prime and $n \in \mathbb{N}$. Prove that the equation has at least $3n + 3$ different solutions $(x,y,z)$ with natural numbers $x,y,z$ and $x < y < z$. Prove the same for $p > 3$ being an odd integer.

PEN H Problems, 67

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 $a, b, c \;$?

2011 Dutch IMO TST, 1

Find all pairs $(x, y)$ of integers that satisfy $x^2 + y^2 + 3^3 = 456\sqrt{x - y}$.

1969 Swedish Mathematical Competition, 1

Find all integers m, n such that $m^3 = n^3 + n$.

2006 QEDMO 2nd, 1

Solve the equation $x^{2}+y^{2}=10xy$ for integers $x$ and $y$

1997 Singapore MO Open, 3

Find all the natural numbers $N$ which satisfy the following properties: (i) $N$ has exactly $6$ distinct factors $1, d_1, d_2, d_3, d_4, N$ and (ii) $1 + N = 5(d_1 + d_2+d_3 + d_4)$. Justify your answers.

2021 Bosnia and Herzegovina Junior BMO TST, 2

Let $p, q, r$ be prime numbers and $t, n$ be natural numbers such that $p^2 +qt =(p + t)^n$ and $p^2 + qr = t^4$ . a) Show that $n < 3$. b) Determine all the numbers $p, q, r, t, n$ that satisfy the given conditions.

2014 Dutch Mathematical Olympiad, 1

Determine all triples $(a,b,c)$, where $a, b$, and $c$ are positive integers that satisfy $a \le b \le c$ and $abc = 2(a + b + c)$.

2009 District Olympiad, 4

Positive integer numbers a and b satisfy $(a^2- 9b^2)^2 - 33b = 1$. a) Prove $|a -3b|\ge 1$. b) Find all pairs of positive integers $(a, b)$ satisfying the equality.

2017 Hanoi Open Mathematics Competitions, 2

How many pairs of positive integers $(x, y)$ are there, those satisfy the identity $2^x - y^2 = 4$? (A): $1$ (B): $2$ (C): $3$ (D): $4$ (E): None of the above.

2021 Korea - Final Round, P2

Positive integer $k(\ge 8)$ is given. Prove that if there exists a pair of positive integers $(x,y)$ that satisfies the conditions below, then there exists infinitely many pairs $(x,y)$. (1) $ $ $x\mid y^2-3, y\mid x^2-2$ (2) $ $ $gcd\left(3x+\frac{2(y^2-3)}{x},2y+\frac{3(x^2-2)}{y}\right)=k$ $ $

2009 Postal Coaching, 2

Find all non-negative integers $a, b, c, d$ such that $7^a = 4^b + 5^c + 6^d$

2016 Saudi Arabia IMO TST, 1

Let $k$ be a positive integer. Prove that there exist integers $x$ and $y$, neither of which divisible by $7$, such that \begin{align*} x^2 + 6y^2 = 7^k. \end{align*}

2016 Latvia Baltic Way TST, 17

Can you find five prime numbers $p, q, r, s, t$ such that $p^3+q^3+r^3+s^3 =t^3$?

1999 Singapore Senior Math Olympiad, 1

Find all the integral solutions of the equation $\left( 1+\frac{1}{x}\right)^{x+1}=\left( 1+\frac{1}{1999}\right)^{1999}$

2022 District Olympiad, P3

$a)$ Solve over the positive integers $3^x=x+2.$ $b)$ Find pairs $(x,y)\in\mathbb{N}\times\mathbb{N}$ such that $(x+3^y)$ and $(y+3^x)$ are consecutive.

2017 Regional Olympiad of Mexico Southeast, 4

Find all couples of positive integers $m$ and $n$ such that $$n!+5=m^3$$

2008 Hanoi Open Mathematics Competitions, 3

Show that the equation $x^2 + 8z = 3 + 2y^2$ has no solutions of positive integers $x, y$ and $z$.