Olympiad problems

2013 Czech-Polish-Slovak Junior Match, 1

Determine all pairs $(x, y)$ of integers for which satisfy the equality $\sqrt{x-\sqrt{y}}+ \sqrt{x+\sqrt{y}}= \sqrt{xy}$

2019 Junior Balkan Team Selection Tests - Romania, 1

Tags: number theory , diophantine , diophantine equation

Determine all positive integers $k$ for which there exist positive integers $n$ and $m, m\ge 2$, such that $3^k + 5^k = n^m$

2014 Swedish Mathematical Competition, 6

Tags: number theory , diophantine , diophantine equation

Determine all odd primes $p$ and $q$ such that the equation $x^p + y^q = pq$ at least one solution $(x, y)$ where $x$ and $y$ are positive integers.

1998 Korea - Final Round, 1

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

Find all pairwise relatively prime positive integers $l, m, n$ such that \[(l+m+n)\left( \frac{1}{l}+\frac{1}{m}+\frac{1}{n}\right)\] is an integer.

2013 JBMO Shortlist, 5

Tags: number theory , diophantine equation , positive integer

Solve in positive integers: $\frac{1}{x^2}+\frac{y}{xz}+\frac{1}{z^2}=\frac{1}{2013}$ .

2021 Miklós Schweitzer, 2

Tags: factorial , diophantine equation , number theory

Prove that the equation \[ 2^x + 5^y - 31^z = n! \] has only a finite number of non-negative integer solutions $x,y,z,n$.

1994 Tournament Of Towns, (413) 1

Tags: number theory , diophantine , diophantine equation

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)

1971 IMO Longlists, 14

Tags: geometry , 3d geometry , diophantine equation , number theory

Note that $8^3 - 7^3 = 169 = 13^2$ and $13 = 2^2 + 3^2.$ Prove that if the difference between two consecutive cubes is a square, then it is the square of the sum of two consecutive squares.

2014 JBMO Shortlist, 2

Tags: modular arithmetic , diophantine equation

Find all triples of primes $(p,q,r)$ satisfying $3p^{4}-5q^{4}-4r^{2}=26$.

PEN H Problems, 68

Tags: ratio , calculus , diophantine equation

Consider the system \[x+y=z+u,\] \[2xy=zu.\] Find the greatest value of the real constant $m$ such that $m \le \frac{x}{y}$ for any positive integer solution $(x, y, z, u)$ of the system, with $x \ge y$.

2016 Korea Winter Program Practice Test, 1

Tags: diophantine equation , number theory

Solve: $a, b, m, n\in \mathbb{N}$ $a^2+b^2=m^2-n^2, ab=2mn$

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 $n$.

2019 Spain Mathematical Olympiad, 4

Tags: number theory , diophantine equation

Find all pairs of integers $(x,y)$ that satisfy the equation $3^4 2^3(x^2+y^2)=x^3y^3$

PEN H Problems, 8

Tags: calculus , integration , diophantine equation

Show that the equation \[x^{3}+y^{3}+z^{3}+t^{3}=1999\] has infinitely many integral solutions.

2017 Israel National Olympiad, 4

Tags: algebra , number theory , diophantine equation

Three rational number $x,p,q$ satisfy $p^2-xq^2$=1. Prove that there are integers $a,b$ such that $p=\frac{a^2+xb^2}{a^2-xb^2}$ and $q=\frac{2ab}{a^2-xb^2}$.

VMEO III 2006, 12.2

Tags: number theory , diophantine equation , diophantine

Find all positive integers $(m, n)$ that satisfy $$m^2 =\sqrt{n} +\sqrt{2n + 1}.$$

2019 Hanoi Open Mathematics Competitions, 11

Tags: number theory , diophantine , diophantine equation

Find all integers $x$ and $y$ satisfying the following equation $x^2 - 2xy + 5y^2 + 2x - 6y - 3 = 0$.

1996 Estonia National Olympiad, 1

Tags: prime , diophantine equation , diophantine , number theory

Let $p$ be a fixed prime. Find all pairs $(x,y)$ of positive numbers satisfying $p(x-y) = xy$.

2018 Thailand Mathematical Olympiad, 6

Tags: diophantine equation , diophantine , number theory

Let $A$ be the set of all triples $(x, y, z)$ of positive integers satisfying $2x^2 + 3y^3 = 4z^4$ . a) Show that if $(x, y, z) \in A$ then $6$ divides all of $x, y, z$. b) Show that $A$ is an infinite set.

2003 All-Russian Olympiad Regional Round, 11.1

Tags: number theory , diophantine , diophantine equation

Find all prime $p$, for each of which there are such natural $ x$ and $y$ such that $p^x = y^3 + 1$.

2021 OMpD, 3

Tags: diophantine equation , number theory

Determine all pairs of integer numbers $(x, y)$ such that: $$\frac{(x - y)^2}{x + y} = x - y + 6$$

PEN H Problems, 45

Tags: diophantine equation

Show that there cannot be four squares in arithmetical progression.

PEN A Problems, 2

Tags: arithmetic sequence , diophantine equation , divisibility theory , pell s equation

Find infinitely many triples $(a, b, c)$ of positive integers such that $a$, $b$, $c$ are in arithmetic progression and such that $ab+1$, $bc+1$, and $ca+1$ are perfect squares.

1998 Croatia National Olympiad, Problem 2

Tags: number theory , diophantine equation

Find all positive integer solutions of the equation $10(m+n)=mn$.

2020 Malaysia IMONST 1, 16

Tags: number theory , diophantine equation

Find the number of positive integer solutions $(a,b,c,d)$ to the equation \[(a^2+b^2)(c^2-d^2)=2020.\] Note: The solutions $(10,1,6,4)$ and $(1,10,6,4)$ are considered different.