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

1997 All-Russian Olympiad Regional Round, 8.7
Tags: diophantine , number theory , diophantine equation
Find all pairs of prime numbers $p$ and $q$ such that $p^3-q^5 = (p+q)^2$.

2010 Grand Duchy of Lithuania, 2
Tags: diophantine , number theory , diophantine equation
Find all positive integers $n$ for which there are distinct integer numbers $a_1, a_2, ... , a_n$ such that $$\frac{1}{a_1}+\frac{2}{a_2}+...+\frac{n}{a_n}=\frac{a_1 + a_2 + ... + a_n}{2}$$

1980 All Soviet Union Mathematical Olympiad, 288
Tags: number theory , diophantine , diophantine equation
Are there three integers $x,y,z$, such that $x^2 + y^3 = z^4$?

2011 NZMOC Camp Selection Problems, 6
Tags: number theory , diophantine equation , diophantine
Find all pairs of non-negative integers $m$ and $n$ that satisfy $$3 \cdot 2^m + 1 = n^2.$$

1990 All Soviet Union Mathematical Olympiad, 523
Tags: floor function , number theory , diophantine
Find all integers $n$ such that $\left[\frac{n}{1!}\right] + \left[\frac{n}{2!}\right] + ... + \left[\frac{n}{10!}\right] = 1001$.

2022 Indonesia TST, N
Tags: quadratic , greatest common divisor , diophantine , integer , number theory
For each natural number $n$, let $f(n)$ denote the number of ordered integer pairs $(x,y)$ satisfying the following equation: \[ x^2 - xy + y^2 = n. \] a) Determine $f(2022)$. b) Determine the largest natural number $m$ such that $m$ divides $f(n)$ for every natural number $n$.

2014 Saudi Arabia GMO TST, 1
Tags: diophantine , number theory , diophantine equation
Find all ordered triples $(a,b, c)$ of positive integers which satisfy $5^a + 3^b - 2^c = 32$

1975 Chisinau City MO, 92
Tags: number theory , diophantine equation , diophantine
Solve in natural numbers the equation $x^2-y^2=105$.

1991 Greece National Olympiad, 4
Tags: number theory , diophantine , diophantine equation
Find all positive intger solutions of $3^x+29=2^y$.

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 $u$.

2014 Belarusian National Olympiad, 2
Tags: number theory , diophantine , diophantine equation
Pairwise distinct prime numbers $p, q, r$ satisfy the equality $$rp^3 + p^2 + p = 2rq^2 +q^2 + q.$$ Determine all possible values of the product $pqr$.

2016 Costa Rica - Final Round, A2
Tags: diophantine , diophantine equation , number theory
Find all integer solutions of the equation $p (x + y) = xy$, where $p$ is a prime number.

2004 Gheorghe Vranceanu, 4
Tags: diophantine , equation
Given a natural prime $ p, $ find the number of integer solutions of the equation $ p+xy=p(x+y). $

2015 QEDMO 14th, 3
Tags: number theory , diophantine , diophantine equation
Are there any rational numbers $x,y$ with $x^2 + y^2 = 2015$?

1981 Polish MO Finals, 5
Tags: diophantine , number theory
Determine all pairs of integers $(x,y)$ satisfying the equation $$x^3 +x^2y+xy^2 +y^3 = 8(x^2 +xy+y^2 +1).$$

2013 Singapore Senior Math Olympiad, 2
Tags: diophantine , diophantine equation , number theory
Find all pairs of integers $(m,n)$ such that $m^3-n^3=2mn +8$

1955 Moscow Mathematical Olympiad, 302
Tags: diophantine , diophantine equation , number theory
Find integer solutions of the equation $x^3 - 2y^3 - 4z^3 = 0$.

2014 Belarus Team Selection Test, 4
Tags: number theory , diophantine , diophantine equation
Find all integers $a$ and $b$ satisfying the equality $3^a - 5^b = 2$. (I. Gorodnin)

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

1994 Abels Math Contest (Norwegian MO), 2a
Tags: number theory , diophantine , diophantine equation , prime
Find all primes $p,q,r$ and natural numbers $n$ such that $\frac{1}{p}+\frac{1}{q}+\frac{1}{r}=\frac{1}{n}$.

2015 Saudi Arabia Pre-TST, 2.3
Tags: number theory , diophantine , diophantine equation
Find all integer solutions of the equation $14^x - 3^y = 2015$. (Malik Talbi)

2007 Nicolae Coculescu, 2
Tags: newton s binom , diophantine , equation
[b]a)[/b] Prove that there exists two infinite sequences $ \left( a_n \right)_{n\ge 1} ,\left( b_n \right)_{n\ge 1} $ of nonnegative integers such that $ a_n>b_n $ and $ (2+\sqrt 3)^n =a_n (2+\sqrt 3) -b_n , $ for any natural numbers $ n. $ [b]b)[/b] Prove that the equation $ x^2-4xy+y^2=1 $ has infinitely many solutions in $ \mathbb{N}^2. $ [i]Florian Dumitrel[/i]

2009 Thailand Mathematical Olympiad, 4
Tags: number theory , diophantine , diophantine equation
Let $k$ be a positive integer. Show that there are infinitely many positive integer solutions $(m, n)$ to $(m - n)^2 = kmn + m + n$.

1945 Moscow Mathematical Olympiad, 103
Tags: number theory , diophantine
Solve in integers the equation $xy + 3x - 5y = - 3$.

2021 Austrian Junior Regional Competition, 4
Tags: number theory , diophantine
Let $p$ be a prime number and let $m$ and $n$ be positive integers with $p^2 + m^2 = n^2$. Prove that $m> p$. (Karl Czakler)