Olympiad problems

2016 Costa Rica - Final Round, N1

Let $p> 5$ be a prime such that none of its digits is divisible by $3$ or $7$. Prove that the equation $x^4 + p = 3y^4$ does not have integer solutions.

1995 Spain Mathematical Olympiad, 4

Tags: diophantine equation , integer solutions , number theory

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

1967 IMO Shortlist, 4

Tags: number theory , perfect cube , diophantine equation , modular arithmetic

Does there exist an integer such that its cube is equal to $3n^2 + 3n + 7,$ where $n$ is an integer.

1946 Moscow Mathematical Olympiad, 117

Tags: diophantine equation , number theory

Prove that for any integers $x$ and $y$ we have $x^5 + 3x^4y - 5x^3y^2 - 15x^2y^3 + 4xy^4 + 12y^5 \ne 33$.

2009 Swedish Mathematical Competition, 4

Tags: diophantine , number theory , diophantine equation

Determine all integers solutions of the equation $x + x^3 = 5y^2$.

1993 Austrian-Polish Competition, 1

Tags: diophantine , diophantine equation , number theory

Solve in positive integers $x,y$ the equation $2^x - 3^y = 7$.

2023 Bangladesh Mathematical Olympiad, P1

Tags: diophantine equation , number theory , factorial

Find all possible non-negative integer solution ($x,$ $y$) of the following equation- $$x!+2^y=z!$$ Note: $x!=x\cdot(x-1)!$ and $0!=1$. For example, $5!=5\times4\times3\times2\times1=120$.

2014 JBMO Shortlist, 2

Tags: diophantine equation , modular arithmetic

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

2021 Ukraine National Mathematical Olympiad, 5

Tags: number theory , diophantine , diophantine equation

Are there natural numbers $(m,n,k)$ that satisfy the equation $m^m+ n^n=k^k$ ?

2007 Moldova National Olympiad, 9.4

Tags: diophantine equation

Find all rational terms of sequence defined by formula $ a_n=\sqrt{\frac{9n-2}{n+1}}, n \in N $

2024 Girls in Mathematics Tournament, 2

Tags: diophantine equation , number theory

Show that there are no triples of positive integers $(x,y,z)$ satisfying the equation \[x^2= 5^y+3^z\]

2018 Belarusian National Olympiad, 10.5

Tags: diophantine equation , number theory

Find all positive integers $n$ such that equation $$3a^2-b^2=2018^n$$ has a solution in integers $a$ and $b$.

2022 German National Olympiad, 4

Tags: cubic equation , diophantine equation , number theory

Determine all $6$-tuples $(x,y,z,u,v,w)$ of integers satisfying the equation \[x^3+7y^3+49z^3=2u^3+14v^3+98w^3.\]

2006 Petru Moroșan-Trident, 2

Tags: algebra , diophantine equation , equation

Solve the following Diophantines. [b]a)[/b] $ x^2+y^2=6z^2 $ [b]b)[/b] $ x^2+y^2-2x+4y-1=0 $ [i]Dan Negulescu[/i]

2016 India PRMO, 1

Tags: diophantine , diophantine equation , algebra

Consider all possible integers $n \ge 0$ such that $(5 \cdot 3^m) + 4 = n^2$ holds for some corresponding integer $m \ge 0$. Find the sum of all such $n$.

PEN H Problems, 25

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

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

2007 Abels Math Contest (Norwegian MO) Final, 3

Tags: number theory , diophantine equation , diophantine

(a) Let $x$ and $y$ be two positive integers such that $\sqrt{x} +\sqrt{y}$ is an integer. Show that $\sqrt{x}$ and $\sqrt{y}$ are both integers. (b) Find all positive integers $x$ and $y$ such that $\sqrt{x} +\sqrt{y}=\sqrt{2007}$.

2012 IMAC Arhimede, 4

Tags: diophantine equation , number theory

Solve the following equations in the set of natural numbers: a) $(5+11\sqrt2)^p=(11+5\sqrt2)^q$ b) $1005^x+2011^y=1006^z$

2011 JBMO Shortlist, 1

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

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

2008 Costa Rica - Final Round, 5

Tags: diophantine equation , other problems

Let $ p$ be a prime number such that $ p\minus{}1$ is a perfect square. Prove that the equation $ a^{2}\plus{}(p\minus{}1)b^{2}\equal{}pc^{2}$ has infinite many integer solutions $ a$, $ b$ and $ c$ with $ (a,b,c)\equal{}1$

2021 New Zealand MO, 5

Tags: diophantine equation , diophantine , number theory

Find all pairs of integers $x, y$ such that $y^5 + 2xy = x^2 + 2y^4.$ .

2007 India IMO Training Camp, 2

Tags: modular arithmetic , diophantine equation , number theory

Find all integer solutions $(x,y)$ of the equation $y^2=x^3-p^2x,$ where $p$ is a prime such that $p\equiv 3 \mod 4.$

2018 Hanoi Open Mathematics Competitions, 15

Tags: diophantine equation , number theory

Find all pairs of prime numbers $(p,q)$ such that for each pair $(p,q)$, there is a positive integer m satisfying $\frac{pq}{p + q}=\frac{m^2 + 6}{m + 1}$.

1999 German National Olympiad, 1

Tags: diophantine equation , diophantine , number theory

Find all $x,y$ which satisfy the equality $x^2 +xy+y^2 = 97$, when $x,y$ are a) natural numbers, b) integers

2013 Iran MO (3rd Round), 4

Tags: polynomial , vieta , diophantine equation , pell equation , number theory , algebra

Prime $p=n^2 +1$ is given. Find the sets of solutions to the below equation: \[x^2 - (n^2 +1)y^2 = n^2.\] (25 points)