Olympiad problems

1978 Yugoslav Team Selection Test, Problem 1

Find all integers $x,y,z$ such that $x^2(x^2+y)=y^{z+1}$.

2009 Croatia Team Selection Test, 4

Tags: number theory , quadratic , diophantine equation

Prove that there are infinite many positive integers $ n$ such that $ n^2\plus{}1\mid n!$, and infinite many of those for which $ n^2\plus{}1 \nmid n!$.

2019 Austrian Junior Regional Competition, 1

Tags: number theory , diophantine equation , diophantine

Let $x$ and $y$ be integers with $x + y \ne 0$. Find all pairs $(x, y)$ such that $$\frac{x^2 + y^2}{x + y}= 10.$$ (Walther Janous)

Russian TST 2016, P1

Tags: diophantine equation , number theory

Find all $ x, y, z\in\mathbb{Z}^+ $ such that \[ (x-y)(y-z)(z-x)=x+y+z \]

2011 Baltic Way, 20

Tags: number theory , diophantine equation

An integer $n\ge 1$ is called balanced if it has an even number of distinct prime divisors. Prove that there exist infinitely many positive integers $n$ such that there are exactly two balanced numbers among $n,n+1,n+2$ and $n+3$.

2023 Romanian Master of Mathematics, 1

Tags: prime number , diophantine equation , number theory

Determine all prime numbers $p$ and all positive integers $x$ and $y$ satisfying $$x^3+y^3=p(xy+p).$$

1969 Dutch Mathematical Olympiad, 2

Tags: number theory , diophantine , diophantine equation

Prove that for all $n \in N$, $x^2 + y^2 = z^n$ has solutions with $x,y,z \in N$.

1980 IMO, 3

Tags: algebra , equation , diophantine equation , number theory

Prove that the equation \[ x^n + 1 = y^{n+1}, \] where $n$ is a positive integer not smaller then 2, has no positive integer solutions in $x$ and $y$ for which $x$ and $n+1$ are relatively prime.

2022 Czech-Polish-Slovak Junior Match, 4

Tags: number theory , diophantine equation , diophantine

Find all triples $(a, b, c)$ of integers that satisfy the equations $ a + b = c$ and $a^2 + b^3 = c^2$

2010 Malaysia National Olympiad, 6

Tags: diophantine equation , algebra

Find the smallest integer $k\ge3$ with the property that it is possible to choose two of the number $1,2,...,k$ in such a way that their product is equal to the sum of the remaining $k-2$ numbers.

2017 Saudi Arabia JBMO TST, 3

Tags: diophantine , diophantine equation , number theory

Find all pairs of primes $(p, q)$ such that $p^3 - q^5 = (p + q)^2$ .

2011 Croatia Team Selection Test, 4

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

Find all pairs of integers $x,y$ for which \[x^3+x^2+x=y^2+y.\]

2013 District Olympiad, 1

Tags: diophantine equation , number theory , diophantine

Find all triples of integers $(x, y, z)$ such that $$x^2 + y^2 + z^2 = 16(x + y + z).$$

1995 India National Olympiad, 2

Tags: number theory , quadratic , diophantine equation , algebra , relatively prime

Show that there are infintely many pairs $(a,b)$ of relatively prime integers (not necessarily positive) such that both the equations \begin{eqnarray*} x^2 +ax +b &=& 0 \\ x^2 + 2ax + b &=& 0 \\ \end{eqnarray*} have integer roots.

2002 Junior Balkan Team Selection Tests - Romania, 1

Tags: diophantine , number theory , diophantine equation

Let $m,n > 1$ be integer numbers. Solve in positive integers $x^n+y^n = 2^m$.

PEN H Problems, 34

Tags: diophantine equation , quadratic

Are there integers $m$ and $n$ such that $5m^2 -6mn+7n^2 =1985$?

2025 Korea - Final Round, P6

Tags: diophantine equation , number theory

Positive integers $a, b$ satisfy both of the following conditions. For a positive integer $m$, if $m^2 \mid ab$, then $m = 1$. There exist integers $x, y, z, w$ that satisfies the equation $ax^2 + by^2 = z^2 + w^2$ and $z^2 + w^2 > 0$. Prove that there exist integers $x, y, z, w$ that satisfies the equation $ax^2 + by^2 + n = z^2 + w^2$, for each integer $n$.

1978 Yugoslav Team Selection Test, Problem 1

2009 Croatia Team Selection Test, 4

2019 Austrian Junior Regional Competition, 1

Russian TST 2016, P1

2011 Baltic Way, 20

2023 Romanian Master of Mathematics, 1

1969 Dutch Mathematical Olympiad, 2

1980 IMO, 3

2022 Czech-Polish-Slovak Junior Match, 4

2010 Malaysia National Olympiad, 6

2017 Saudi Arabia JBMO TST, 3

2011 Croatia Team Selection Test, 4

2013 District Olympiad, 1

1995 India National Olympiad, 2

2002 Junior Balkan Team Selection Tests - Romania, 1

PEN H Problems, 34

2025 Korea - Final Round, P6

2019 Kosovo National Mathematical Olympiad, 5

2022 Abelkonkurransen Finale, 1b

2002 Croatia National Olympiad, Problem 4

PEN H Problems, 32

2009 Croatia Team Selection Test, 4

2014 Turkey EGMO TST, 2

2009 May Olympiad, 2

1984 Tournament Of Towns, (069) T3