Olympiad problems

2007 Argentina National Olympiad, 1

Find all the prime numbers $p$ and $q$ such that $ p^2+q=37q^2+p $. Clarification: $1$ is not a prime number.

1984 Poland - Second Round, 1

Tags: number theory , diophantine , diophantine equation

For a given natural number $ n $, find the number of solutions to the equation $ \sqrt{x} + \sqrt{y} = n $ in natural numbers $ x, y $.

2010 Grand Duchy of Lithuania, 2

Tags: number theory , diophantine , diophantine equation

Find all positive integers $n$ for which there are distinct integer numbers $a_1, a_2, ... , a_n$ such that $$\frac{1}{a_1}+\frac{2}{a_2}+...+\frac{n}{a_n}=\frac{a_1 + a_2 + ... + a_n}{2}$$

2018 Pan-African Shortlist, N7

Tags: diophantine equation , number theory

Find all non-negative integers $n$ for which the equation \[ {\left( x^2 + y^2 \right)}^n = {(xy)}^{2018} \] admits positive integral solutions.

2009 VTRMC, Problem 6

Tags: number theory , diophantine equation

Let $n$ be a nonzero integer. Prove that $n^4-7n^2+1$ can never be a perfect square.

PEN H Problems, 60

Tags: diophantine equation , modular arithmetic

Show that the equation $x^7 + y^7 = {1998}^z$ has no solution in positive integers.

2012 Dutch IMO TST, 3

Tags: diophantine equation , diophantine , number theory

Determine all pairs $(x, y)$ of positive integers satisfying $x + y + 1 | 2xy$ and $ x + y - 1 | x^2 + y^2 - 1$.

1982 Austrian-Polish Competition, 7

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

Find the triple of positive integers $(x,y,z)$ with $z$ least possible for which there are positive integers $a, b, c, d$ with the following properties: (i) $x^y = a^b = c^d$ and $x > a > c$ (ii) $z = ab = cd$ (iii) $x + y = a + b$.

2008 Korea Junior Math Olympiad, 3

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

For all positive integers $n$, prove that there are integers $x, y$ relatively prime to $5$ such that $x^2 + y^2 = 5^n$.

2015 Indonesia MO Shortlist, N4

Tags: number theory , exponential equation , exponential , greatest common divisor , diophantine equation

Suppose that the natural number $a, b, c, d$ satisfy the equation $a^ab^{a + b} = c^cd^{c + d}$. (a) If gcd $(a, b) = $ gcd $(c, d) = 1$, prove that $a = c$ and $b = d$. (b) Does the conclusion $a = c$ and $b = d$ apply, without the condition gcd $(a, b) = $ gcd $(c, d) = 1$?

2002 Croatia National Olympiad, Problem 4

Tags: number theory , diophantine equation

Find all natural numbers $n$ for which the equation $\frac1x+\frac1y=\frac1n$ has exactly five solutions $(x,y)$ in the set of natural numbers.

2011 Greece JBMO TST, 3

Tags: integer , number theory , diophantine equation

Find integer solutions of the equation $8x^3 - 4 = y(6x - y^2)$

2023 German National Olympiad, 1

Tags: diophantine equation , number theory , algebra

Determine all pairs $(m,n)$ of integers with $n \ge m$ satisfying the equation \[n^3+m^3-nm(n+m)=2023.\]

2021 Serbia Team Selection Test, P3

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

Given is a prime number $p$. Find the number of positive integer solutions $(a, b, c, d)$ of the system of equations $ac+bd = p(a+c)$ and $bc-ad = p(b-d)$.

2013 Junior Balkan Team Selection Tests - Romania, 1

Tags: diophantine , diophantine equation , number theory

Let $n$ be a positive integer. Determine all positive integers $p$ for which there exist positive integers $x_1 < x_2 <...< x_n$ such that $\frac{1}{x_1}+\frac{2}{x_2}+ ... +\frac{n}{x_n}= p$ Irish Mathematical Olympiad

2016 Abels Math Contest (Norwegian MO) Final, 2b

Tags: factorial , diophantine equation , diophantine , number theory

Find all non-negative integers $x, y$ and $z$ such that $x^3 + 2y^3 + 4z^3 = 9!$

2018 JBMO Shortlist, NT2

Tags: number theory , diophantine equation

Find all ordered pairs of positive integers $(m,n)$ such that : $125*2^n-3^m=271$

2012 France Team Selection Test, 3

Tags: modular arithmetic , number theory , diophantine equation

Let $p$ be a prime number. Find all positive integers $a,b,c\ge 1$ such that: \[a^p+b^p=p^c.\]

2005 Czech-Polish-Slovak Match, 6

Tags: modular arithmetic , diophantine equation , prime factorization , number theory

Determine all pairs of integers $(x, y)$ satisfying the equation \[y(x + y) = x^3- 7x^2 + 11x - 3.\]

2016 Balkan MO Shortlist, N3

Tags: number theory , diophantine equation , diophantine

Find all the integer solutions $(x,y,z)$ of the equation $(x + y + z)^5 = 80xyz(x^2 + y^2 + z^2)$,

1979 Bulgaria National Olympiad, Problem 1

Tags: number theory , diophantine equation , diophantine

Show that there are no integers $x$ and $y$ satisfying $x^2 + 5 = y^3$. Daniel Harrer

2000 Estonia National Olympiad, 3

Tags: number theory , diophantine , diophantine equation

Are there any (not necessarily positive) integers $m$ and $n$ such that a) $\frac{1}{m}-\frac{1}{n}=\frac{1}{m-n}$ ? b) $\frac{1}{m}-\frac{1}{n}=\frac{1}{n-m}$

2023 Singapore Senior Math Olympiad, 2

Tags: diophantine equation , number theory

Find all positive integers $k$ such that there exists positive integers $a, b$ such that \[a^2+4=(k^2-4)b^2.\]

2019 Peru EGMO TST, 1

Tags: diophantine equation , diophantine , number theory , prime

Find all the prime numbers $p, q$ and $r$ such that $p^2 + 1 = 74 (q^2 + r^2)$.

PEN H Problems, 25

Tags: diophantine equation , geometry , 3d geometry , modular arithmetic

What is the smallest positive integer $t$ such that there exist integers $x_{1},x_{2}, \cdots, x_{t}$ with \[{x_{1}}^{3}+{x_{2}}^{3}+\cdots+{x_{t}}^{3}=2002^{2002}\;\;?\]