Olympiad problems

1980 All Soviet Union Mathematical Olympiad, 288

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

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 $a^2+ab+b^2=c^2+3$.

2013 District Olympiad, 1

Tags: algebra , diophantine

Prove that the equation $$\frac{1}{\sqrt{x} +\sqrt{1006}}+\frac{1}{\sqrt{2012 -x} +\sqrt{1006}}=\frac{2}{\sqrt{x} +\sqrt{2012 -x}}$$ has $2013$ integer solutions.

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.

2009 Postal Coaching, 2

Tags: diophantine , number theory , diophantine equation

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

2015 Canadian Mathematical Olympiad Qualification, 1

Tags: number theory , diophantine

Find all integer solutions to the equation $7x^2y^2 + 4x^2 = 77y^2 + 1260$.

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 $$5x + 2y + z = 10n.$$

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 $u$.

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 $(x , y , z )$ : $\begin{cases}x + y = a - 1 \\ x(y + 1) - z^2 = b \end{cases}$

2022 Czech-Polish-Slovak Junior Match, 2

Tags: diophantine equation , number theory , diophantine

Solve the following system of equations in integer numbers: $$\begin{cases} x^2 = yz + 1 \\ y^2 = zx + 1 \\ z^2 = xy + 1 \end{cases}$$

2017 Saudi Arabia BMO TST, 2

Tags: number theory , diophantine , diophantine equation

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

2018 NZMOC Camp Selection Problems, 9

Tags: diophantine , diophantine equation , number theory , power

Let $x, y, p, n, k$ be positive integers such that $$x^n + y^n = p^k.$$ Prove that if $n > 1$ is odd, and $p$ is an odd prime, then $n$ is a power of $p$.

2014 Junior Balkan Team Selection Tests - Romania, 2

Tags: diophantine , diophantine equation , number theory

Solve, in the positive integers, the equation $5^m + n^2 = 3^p$ .

2005 Switzerland - Final Round, 7

Tags: diophantine , diophantine equation , number theory

Let $n\ge 1$ be a natural number. Determine all positive integer solutions of the equation $$7 \cdot 4^n = a^2 + b^2 + c^2 + d^2.$$

2005 Thailand Mathematical Olympiad, 13

Tags: diophantine , diophantine equation , number theory , odd

Find all odd integers $k$ for which there exists a positive integer $m$ satisfying the equation $k + (k + 5) + (k + 10) + ... + (k + 5(m - 1)) = 1372$.

2010 China Northern MO, 3

Tags: number theory , diophantine equation , diophantine

Find all positive integer triples $(x, y, z)$ such that $1 + 2^x \cdot 3^y=5^z$ is true.

2022 New Zealand MO, 1

Tags: number theory , diophantine equation , diophantine

Find all integers $a, b$ such that $$a^2 + b = b^{2022}.$$

2018 IMO Shortlist, N5

Tags: diophantine , number theory

Four positive integers $x,y,z$ and $t$ satisfy the relations \[ xy - zt = x + y = z + t. \] Is it possible that both $xy$ and $zt$ are perfect squares?

2011 Grand Duchy of Lithuania, 3

Tags: diophantine , diophantine equation , number theory

Find all primes $p,q$ such that $p ^3-q^7=p-q$.

2006 Bosnia and Herzegovina Junior BMO TST, 1

Tags: number theory , diophantine , diophantine equation

. Find all triplets $(x, y, z)$, $x > y > z$ of positive integers such that $\frac{1}{x}+\frac{2}{y}+\frac{3}{z}= 1$

2022 Dutch IMO TST, 1

Tags: number theory , diophantine equation , diophantine

Find all quadruples $(a, b, c, d)$ of non-negative integers such that $ab =2(1 + cd)$ and there exists a non-degenerate triangle with sides of length $a - c$, $b - d$, and $c + d$.

2018 NZMOC Camp Selection Problems, 2

Tags: number theory , diophantine , diophantine equation

Find all pairs of integers $(a, b)$ such that $$a^2 + ab - b = 2018.$$