Olympiad problems

2022 Dutch IMO TST, 1

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 $c + d$.

PEN H Problems, 49

Tags: diophantine equation , search , group theory , abstract algebra

Show that the only solutions of the equation $x^{3}-3xy^2 -y^3 =1$ are given by $(x,y)=(1,0),(0,-1),(-1,1),(1,-3),(-3,2),(2,1)$.

2022 Austrian MO National Competition, 4

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

Find all triples $(p, q, r)$ of prime numbers for which $4q - 1$ is a prime number and $$\frac{p + q}{p + r} = r - p$$ holds. [i](Walther Janous)[/i]

2009 Korea National Olympiad, 3

Tags: diophantine equation , number theory

Let $n$ be a positive integer. Suppose that the diophantine equation \[z^n = 8 x^{2009} + 23 y^{2009} \] uniquely has an integer solution $(x,y,z)=(0,0,0)$. Find the possible minimum value of $n$.

2021 Nigerian MO Round 3, Problem 1

Tags: number theory , diophantine equation , prime

Find all triples of primes $(p, q, r)$ such that $p^q=2021+r^3$.

2020 Regional Competition For Advanced Students, 4

Tags: number theory , diophantine equation , diophantine

Find all quadruples $(p, q, r, n)$ of prime numbers $p, q, r$ and positive integer numbers $n$, such that $$p^2 = q^2 + r^n$$ (Walther Janous)

KoMaL A Problems 2019/2020, A. 778

Tags: diophantine equation , number theory

Find all square-free integers $d$ for which there exist positive integers $x, y$ and $n$ satisfying $x^2+dy^2=2^n$ Submitted by Kada Williams, Cambridge

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)

2023 Dutch BxMO TST, 5

Tags: modular arithmetic , diophantine equation , number theory

Find all pairs of prime numbers $(p,q)$ for which \[2^p = 2^{q-2} + q!.\]

1992 Poland - First Round, 1

Tags: number theory , diophantine equation

Solve the following equation in real numbers: $\frac{(x^2-1)(|x|+1)}{x+sgnx}=[x+1].$

2019 Regional Olympiad of Mexico West, 5

Tags: number theory , diophantine equation

Prove that for every integer $n > 1$ there exist integers $x$ and $y$ such that $$\frac{1}{n}=\frac{1}{x(x+1)}+\frac{1}{(x+1)(x+2)}+...+\frac{1}{y(y+1)}.$$

2015 BAMO, 3

Tags: diophantine equation , algebra

Let $k$ be a positive integer. Prove that there exist integers $x$ and $y$, neither of which is divisible by $3$, such that $x^2+2y^2 = 3^k$.

2017 Istmo Centroamericano MO, 3

Tags: number theory , diophantine equation , diophantine

Find all ordered pairs of integers $(x, y)$ with $y \ge 0$ such that $x^2 + 2xy + y! = 131$.

2021 Grand Duchy of Lithuania, 4

Tags: number theory , diophantine equation , diophantine

A triplet of positive integers $(x, y, z)$ satisfying $x, y, z > 1$ and $x^3 - yz^3 = 2021$ is called [i]primary [/i] if at least two of the integers $x, y, z$ are prime numbers. a) Find at least one primary triplet. b) Show that there are infinitely many primary triplets.

2022 German National Olympiad, 4

Tags: diophantine equation , cubic equation , number theory

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

1991 Chile National Olympiad, 1

Tags: diophantine , diophantine equation , number theory

Determine all nonnegative integer solutions of the equation $2^x-2^y = 1$

PEN H Problems, 20

Tags: diophantine equation

Determine all positive integers $n$ for which the equation \[x^{n}+(2+x)^{n}+(2-x)^{n}= 0\] has an integer as a solution.

PEN H Problems, 82

Tags: diophantine equation

Find all triples $(a, b, c)$ of positive integers to the equation \[a! b! = a!+b!+c!.\]

PEN H Problems, 70

Tags: diophantine equation

Show that the equation $\{x^3\}+\{y^3\}=\{z^3\}$ has infinitely many rational non-integer solutions.

PEN H Problems, 69

Tags: diophantine equation , number theory , relatively prime

Determine all positive rational numbers $r \neq 1$ such that $\sqrt[r-1]{r}$ is rational.

2011 Dutch IMO TST, 1

Tags: diophantine equation , number theory

Find all pairs $(x, y)$ of integers that satisfy $x^2 + y^2 + 3^3 = 456\sqrt{x - y}$.

1999 ITAMO, 6

Tags: diophantine equation , number theory , diophantine

(a) Find all pairs $(x,k)$ of positive integers such that $3^k -1 = x^3$ . (b) Prove that if $n > 1$ is an integer, $n \ne 3$, then there are no pairs $(x,k)$ of positive integers such that $3^k -1 = x^n$.

1979 IMO Longlists, 56

Tags: function , floor function , diophantine equation , algebra

Show that for every $n\in\mathbb{N}$, $n\sqrt{2}-\lfloor n\sqrt{2}\rfloor>\frac{1}{2n \sqrt{2}}$ and that for every $\epsilon >0$, there exists an $n\in\mathbb{N}$ such that $ n\sqrt{2}-\lfloor n\sqrt{2}\rfloor < \frac{1}{2n \sqrt{2}}+\epsilon$.

2014 Thailand TSTST, 2

Tags: number theory , diophantine equation

Prove that the equation $x^8 = n! + 1$ has finitely many solutions in positive integers.

2014 Hanoi Open Mathematics Competitions, 10

Tags: number theory , diophantine equation , diophantine

Find all pairs of integers $(x, y)$ satisfying the condition $12x^2 + 6xy + 3y^2 = 28(x + y)$.