Olympiad problems

2022 Czech-Polish-Slovak Junior Match, 2

Solve the following system of equations in integer numbers: $$\begin{cases} x^2 = yz + 1 \\ y^2 = zx + 1 \\ z^2 = xy + 1 \end{cases}$$

2013 Junior Balkan Team Selection Tests - Moldova, 6

Tags: diophantine , diophantine equation , number theory

Determine all triplets of real numbers $(x, y, z)$ that satisfy the equation $4xyz = x^4 + y^4 + z^4 + 1$.

1991 Swedish Mathematical Competition, 1

Tags: diophantine , diophantine equation , number theory

Find all positive integers $m, n$ such that $\frac{1}{m} + \frac{1}{n} - \frac{1}{mn} =\frac{2}{5}$.

2013 Danube Mathematical Competition, 3

Tags: exponential , number theory , diophantine , diophantine equation , exponential equation

Determine the natural numbers $m,n$ such as $85^m-n^4=4$

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 \[ \frac{m}{n}+\frac{1}{p^2}=\frac{m+p}{n+p} \]

2019 Romania National Olympiad, 4

Tags: diophantine , diophantine equation , number theory

Find the natural numbers $x, y, z$ that verify the equation: $$2^x + 3 \cdot 11^y =7^z$$

2006 Singapore Junior Math Olympiad, 1

Tags: number theory , diophantine , diophantine equation

Find all integers $x,y$ that satisfy the equation $x+y=x^2-xy+y^2$

1967 Swedish Mathematical Competition, 3

Tags: diophantine , diophantine equation , number theory

Show that there are only finitely many triples $(a, b, c)$ of positive integers such that $\frac{1}{a} + \frac{1}{b} + \frac{1}{c} = \frac{1}{1000}$.

2008 Hanoi Open Mathematics Competitions, 4

Tags: diophantine , diophantine equation , number theory

Find all pairs $(m,n)$ of positive integers such that $m^2 + n^2 = 3(m + n)$.

2019 Durer Math Competition Finals, 15

Tags: diophantine , diophantine equation , sum , number theory

The positive integer $m$ and non-negative integers $x_0, x_1,..., x_{1001}$ satisfy the following equation: $$m^{x_0} =\sum_{i=1}^{1001}m^{x_i}.$$ How many possibilities are there for the value of $m$?

2017 Greece Junior Math Olympiad, 3

Tags: diophantine , diophantine equation , number theory

Find all triplets $(a,b,p)$ where $a,b$ are positive integers and $p$ is a prime number satisfying: $\frac{1}{p}=\frac{1}{a^2}+\frac{1}{b^2}$

2001 Croatia Team Selection Test, 3

Tags: diophantine equation , number theory , diophantine

Find all solutions of the equation $(a^a)^5 = b^b$ in positive integers.

1998 Swedish Mathematical Competition, 1

Tags: diophantine , diophantine equation , number theory

Find all positive integers $a, b, c$, such that $(8a-5b)^2 + (3b-2c)^2 + (3c-7a)^2 = 2$.

2016 Costa Rica - Final Round, N1

Tags: diophantine , diophantine equation , number theory

Let $p> 5$ be a prime such that none of its digits is divisible by $3$ or $7$. Prove that the equation $x^4 + p = 3y^4$ does not have integer solutions.

2009 Swedish Mathematical Competition, 4

Tags: diophantine , number theory , diophantine equation

Determine all integers solutions of the equation $x + x^3 = 5y^2$.

1993 Austrian-Polish Competition, 1

Tags: diophantine , diophantine equation , number theory

Solve in positive integers $x,y$ the equation $2^x - 3^y = 7$.

2003 Junior Balkan Team Selection Tests - Romania, 3

Tags: diophantine , radical , number theory , inequalities

Let $n$ be a positive integer. Prove that there are no positive integers $x$ and $y$ such as $\sqrt{n}+\sqrt{n+1} < \sqrt{x}+\sqrt{y} <\sqrt{4n+2} $

2021 Ukraine National Mathematical Olympiad, 5

Tags: number theory , diophantine , diophantine equation

Are there natural numbers $(m,n,k)$ that satisfy the equation $m^m+ n^n=k^k$ ?

2020 JBMO Shortlist, 6

Tags: diophantine , shortlist , number theory

Are there any positive integers $m$ and $n$ satisfying the equation $m^3 = 9n^4 + 170n^2 + 289$ ?

2016 India PRMO, 1

Tags: diophantine , diophantine equation , algebra

Consider all possible integers $n \ge 0$ such that $(5 \cdot 3^m) + 4 = n^2$ holds for some corresponding integer $m \ge 0$. Find the sum of all such $n$.

2007 Abels Math Contest (Norwegian MO) Final, 3

Tags: number theory , diophantine equation , diophantine

(a) Let $x$ and $y$ be two positive integers such that $\sqrt{x} +\sqrt{y}$ is an integer. Show that $\sqrt{x}$ and $\sqrt{y}$ are both integers. (b) Find all positive integers $x$ and $y$ such that $\sqrt{x} +\sqrt{y}=\sqrt{2007}$.

2021 New Zealand MO, 5

Tags: diophantine equation , diophantine , number theory

Find all pairs of integers $x, y$ such that $y^5 + 2xy = x^2 + 2y^4.$ .

1999 German National Olympiad, 1

Tags: diophantine equation , diophantine , number theory

Find all $x,y$ which satisfy the equality $x^2 +xy+y^2 = 97$, when $x,y$ are a) natural numbers, b) integers

2015 NZMOC Camp Selection Problems, 4

Tags: diophantine , diophantine equation , number theory

For which positive integers $m$ does the equation: $$(ab)^{2015} = (a^2 + b^2)^m$$ have positive integer solutions?

2007 QEDMO 4th, 1

Tags: diophantine equation , diophantine , number theory

Find all primes $p,$ $q,$ $r$ satisfying $p^{2}+2q^{2}=r^{2}.$