Olympiad problems

2014 China Northern MO, 3

Determine whether there exist an infinite number of positive integers $x,y $ satisfying the condition: $x^2+y \mid x+y^2.$ Please prove it.

1998 German National Olympiad, 3

Tags: diophantine , diophantine equation , number theory

For each nonnegative integer $k$ find all nonnegative integers $x,y,z$ such that $x^2 +y^2 +z^2 = 8^k$

1994 Tournament Of Towns, (413) 1

Tags: diophantine , diophantine equation , number theory

Does there exist an infinite set of triples of integers $x, y, z$ (not necessarily positive) such that $$x^2 + y^2 + z^2 = x^3 + y^3+z^3?$$ (NB Vassiliev)

2016 Finnish National High School Mathematics Comp, 4

Tags: number theory , diophantine equation , diophantine

How many pairs $(a, b)$ of positive integers $a,b$ solutions of the equation $(4a-b)(4b-a )=1770^n$ exist , if $n$ is a positive integer?

1994 Swedish Mathematical Competition, 4

Tags: number theory , diophantine , diophantine equation

Find all integers $m, n$ such that $2n^3 - m^3 = mn^2 + 11$.

2000 Kazakhstan National Olympiad, 4

Tags: number theory , diophantine equation , diophantine

Find all triples of natural numbers $ (x, y, z) $ that satisfy the condition $ (x + 1) ^ {y + 1} + 1 = (x + 2) ^ {z + 1}. $

2018 Bosnia and Herzegovina Junior BMO TST, 2

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

Find all integer triples $(p,m,n)$ that satisfy: $p^m-n^3=27$ where $p$ is a prime number.

2014 China Team Selection Test, 3

Tags: diophantine equation , number theory

Show that there are no 2-tuples $ (x,y)$ of positive integers satisfying the equation $ (x+1) (x+2)\cdots (x+2014)= (y+1) (y+2)\cdots (y+4028).$

2021 Israel TST, 4

Tags: number theory , diophantine equation

Let $r$ be a positive integer and let $a_r$ be the number of solutions to the equation $3^x-2^y=r$ ,such that $0\leq x,y\leq 5781$ are integers. What is the maximal value of $a_r$?

2024 Mozambique National Olympiad, P5

Tags: algebra , number theory , diophantine equation

Find all pairs of positive integers $x,y$ such that $\frac{4}{x}+\frac{2}{y}=1$

2019 Danube Mathematical Competition, 1

Tags: algebra , diophantine equation , equation

Solve in $ \mathbb{Z}^2 $ the equation: $ x^2\left( 1+x^2 \right) =-1+21^y. $ [i]Lucian Petrescu[/i]

2009 Belarus Team Selection Test, 4

Tags: number theory , perfect square , diophantine , diophantine equation

Let $x,y,z$ be integer numbers satisfying the equality $yx^2+(y^2-z^2)x+y(y-z)^2=0$ a) Prove that number $xy$ is a perfect square. b) Prove that there are infinitely many triples $(x,y,z)$ satisfying the equality. I.Voronovich

2017 Saudi Arabia BMO TST, 2

Tags: diophantine , diophantine equation , number theory

Solve the following equation in positive integers $x, y$: $x^{2017} - 1 = (x - 1)(y^{2015}- 1)$

2013 Greece National Olympiad, 2

Tags: quadratic , diophantine equation , number theory

Solve in integers the following equation: \[y=2x^2+5xy+3y^2\]

2022 China Team Selection Test, 4

Tags: number theory , diophantine equation

Find all positive integers $a,b,c$ and prime $p$ satisfying that \[ 2^a p^b=(p+2)^c+1.\]

2017 Singapore Senior Math Olympiad, 1

Tags: number theory , diophantine , diophantine equation , factorial

Let $n$ be a positive integer and $a_1,a_2,...,a_{2n}$ be $2n$ distinct integers. Given that the equation $|x-a_1| |x-a_2| ... |x-a_{2n}| =(n!)^2$ has an integer solution $x = m$, find $m$ in terms of $a_1,a_2,...,a_{2n}$