Olympiad problems

1995 Austrian-Polish Competition, 7

Consider the equation $3y^4 + 4cy^3 + 2xy + 48 = 0$, where $c$ is an integer parameter. Determine all values of $c$ for which the number of integral solutions $(x,y)$ satisfying the conditions (i) and (ii) is maximal: (i) $|x|$ is a square of an integer; (ii) $y$ is a squarefree number.

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}. $

1992 Swedish Mathematical Competition, 4

Tags: inequalities , number theory , diophantine

Find all positive integers $a, b, c$ such that $a < b$, $a < 4c$, and $b c^3 \le a c^3 + b$.

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 $x, y$: $x^{2017} - 1 = (x - 1)(y^{2015}- 1)$

2004 Austria Beginners' Competition, 2

Tags: inequalities , diophantine , number theory

For what pairs of integers $(x,y)$ does the inequality $x^2+5y^2-6\leq \sqrt{(x^2-2)(y^2-0.04)}$ hold?

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 $a_1,a_2,...,a_{2n}$