Olympiad problems

2014 Estonia Team Selection Test, 6

Find all natural numbers $n$ such that the equation $x^2 + y^2 + z^2 = nxyz$ has solutions in positive integers

2011 JBMO Shortlist, 1

Tags: number theory , algebra , integer equation , diophantine equation , modular arithmetic

Solve in positive integers the equation $1005^x + 2011^y = 1006^z$.

1980 All Soviet Union Mathematical Olympiad, 288

Tags: diophantine , diophantine equation , number theory

Are there three integers $x,y,z$, such that $x^2 + y^3 = z^4$?

2014 Stars Of Mathematics, 1

Tags: number theory , diophantine equation

Prove there exist infinitely many pairs $(x,y)$ of integers $1<x<y$, such that $x^3+y \mid x+y^3$. ([i]Dan Schwarz[/i])

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.

2012 Czech-Polish-Slovak Junior Match, 2

Tags: diophantine equation , diophantine , number theory , prime

Determine all three primes $(a, b, c)$ that satisfied the equality $a^2+ab+b^2=c^2+3$.

1996 Israel National Olympiad, 1

Tags: diophantine , diophantine equation , number theory

Let $a$ be a prime number and $n > 2$ an integer. Find all integer solutions of the equation $x^n +ay^n = a^2z^n$ .

2017 Hanoi Open Mathematics Competitions, 2

Tags: number theory , diophantine , diophantine equation

How many pairs of positive integers $(x, y)$ are there, those satisfy the identity $2^x - y^2 = 1$? (A): $1$ (B): $2$ (C): $3$ (D): $4$ (E): None of the above.

2012 India PRMO, 19

Tags: diophantine equation , number theory

How many integer pairs $(x,y)$ satisfy $x^2+4y^2-2xy-2x-4y-8=0$?

1967 IMO Longlists, 15

Tags: trigonometry , number theory , diophantine equation , trigonometric equation

Suppose $\tan \alpha = \dfrac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$. Prove that the number $\tan \beta$ for which $\tan {2 \beta} = \tan {3 \alpha}$ is rational only when $p^2 + q^2$ is the square of an integer.

2009 Postal Coaching, 2

Tags: diophantine , number theory , diophantine equation

Find all non-negative integers $a, b, c, d$ such that $7^a = 4^b + 5^c + 6^d$

1978 Yugoslav Team Selection Test, Problem 1

Tags: diophantine equation , number theory

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

2011 Croatia Team Selection Test, 4

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

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

2021 Korea - Final Round, P2

Tags: diophantine equation , number theory

Positive integer $k(\ge 8)$ is given. Prove that if there exists a pair of positive integers $(x,y)$ that satisfies the conditions below, then there exists infinitely many pairs $(x,y)$. (1) $ $ $x\mid y^2-3, y\mid x^2-2$ (2) $ $ $gcd\left(3x+\frac{2(y^2-3)}{x},2y+\frac{3(x^2-2)}{y}\right)=k$ $ $

2012 Ukraine Team Selection Test, 7

Tags: relatively prime , number theory , diophantine equation

Find all pairs of relatively prime integers $(x, y)$ that satisfy equality $2 (x^3 - x) = 5 (y^3 - y)$.

1998 Bundeswettbewerb Mathematik, 1

Tags: diophantine equation , diophantine , number theory

Find all integer solutions $(x,y,z)$ of the equation $xy+yz+zx-xyz = 2$.

1967 German National Olympiad, 5

Tags: number theory , diophantine , diophantine equation

For each natural number $n$, determine the number $A(n)$ of all integer nonnegative solutions the equation $$5x + 2y + z = 10n.$$

PEN H Problems, 14

Tags: diophantine equation

Show that the equation $x^2 +y^5 =z^3$ has infinitely many solutions in integers $x, y, z$ for which $xyz \neq 0$.

1995 Spain Mathematical Olympiad, 4

Tags: number theory , diophantine equation , integer solutions

Given a prime number $p$, find all integer solutions of $p(x+y) = xy$.

1991 Bundeswettbewerb Mathematik, 1

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

Determine all solutions of the equation $4^x + 4^y + 4^z = u^2$ for integers $x,y,z$ and $u$.

PEN H Problems, 89

Tags: diophantine equation

Prove that the number $99999+111111\sqrt{3}$ cannot be written in the form $(A+B\sqrt{3})^2$, where $A$ and $B$ are integers.

2017 Hanoi Open Mathematics Competitions, 6

Tags: diophantine equation , diophantine , system of equations , number theory , parametric equation

Find all pairs of integers $a, b$ such that the following system of equations has a unique integral solution $(x , y , z )$ : $\begin{cases}x + y = a - 1 \\ x(y + 1) - z^2 = b \end{cases}$

1998 Junior Balkan Team Selection Tests - Romania, 1

Tags: equation , algebra , diophantine equation

Solve in $ \mathbb{Z}^2 $ the following equation: $$ (x+1)(x+2)(x+3) +x(x+2)(x+3)+x(x+1)(x+3)+x(x+1)(x+2)=y^{2^x} . $$ [i]Adrian Zanoschi[/i]

2022 Israel National Olympiad, P4

Tags: number theory , diophantine equation

Find all triples $(a,b,c)$ of integers for which the equation \[x^3-a^2x^2+b^2x-ab+3c=0\] has three distinct integer roots $x_1,x_2,x_3$ which are pairwise coprime.

2000 Putnam, 2

Tags: quadratic , algebra , polynomial , induction , diophantine equation

Prove that there exist infinitely many integers $n$ such that $n$, $n+1$, $n+2$ are each the sum of the squares of two integers. [Example: $0=0^2+0^2$, $1=0^2+1^2$, $2=1^2+1^2$.]