Olympiad problems

1993 Swedish Mathematical Competition, 3

Assume that $a$ and $b$ are integers. Prove that the equation $a^2 +b^2 +x^2 = y^2$ has an integer solution $x,y$ if and only if the product $ab$ is even.

1973 Swedish Mathematical Competition, 4

Tags: diophantine , number theory , diophantine equation

$p$ is a prime. Find all relatively prime positive integers $m$, $n$ such that \[ \frac{m}{n}+\frac{1}{p^2}=\frac{m+p}{n+p} \]

2020 Francophone Mathematical Olympiad, 4

Tags: number theory , diophantine equation , diophantine

Find all the integers $x, y$ and $z$ greater than or equal to $0$ such that $2^x + 9 \cdot 7^y = z^3$

1998 Poland - Second Round, 4

Tags: diophantine equation , diophantine , number theory

Find all pairs of integers $(x,y)$ satisfying $x^2 +3y^2 = 1998x$.

2015 Costa Rica - Final Round, N1

Tags: number theory , diophantine , diophantine equation

Find all the values of $n \in N$ such that $n^2 = 2^n$.

2003 Singapore MO Open, 3

Tags: diophantine , diophantine equation , number theory

For any given prime $p$, determine whether the equation $x^2 + y^2 + p^z = 2003$ always has integer solutions in $x, y, z$. Justify your answer

2008 Denmark MO - Mohr Contest, 2

Tags: number theory , diophantine , diophantine equation

If three integers $p, q$ and $r$ apply that $$p + q^2 = r ^2.$$Show that $6$ adds up to $pqr$ .

2021 Austrian Junior Regional Competition, 4

Tags: diophantine , number theory

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)

2011 Denmark MO - Mohr Contest, 3

Tags: diophantine , number theory

Determine all the ways in which the fraction $\frac{1}{11}$ can be written as $\frac{1}{n}+\frac{1}{m}$ , where $n$ and $m$ are two different positive integers.

1990 Poland - Second Round, 1

Tags: number theory , diophantine , diophantine equation

Find all pairs of integers $ x $, $ y $ satisfying the equation $$ (xy-1)^2 = (x +1)^2 + (y +1)^2.$$

2018 India PRMO, 18

Tags: diophantine equation , diophantine , number theory

If $a, b, c \ge 4$ are integers, not all equal, and $4abc = (a+3)(b+3)(c+3)$ then what is the value of $a+b+c$ ?