Olympiad problems

1994 Swedish Mathematical Competition, 4

Find all integers $m, n$ such that $2n^3 - m^3 = mn^2 + 11$.

PEN H Problems, 35

Tags: diophantine equation , algebra , polynomial

Find all cubic polynomials $x^3 +ax^2 +bx+c$ admitting the rational numbers $a$, $b$ and $c$ as roots.

2013 Costa Rica - Final Round, N1

Tags: number theory , diophantine equation , diophantine

Find all triples $(a, b, p)$ of positive integers, where $p$ is a prime number, such that $a^p - b^p = 2013$.

1969 IMO Shortlist, 7

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

$(BUL 1)$ Prove that the equation $\sqrt{x^3 + y^3 + z^3}=1969$ has no integral solutions.

2015 Junior Balkan Team Selection Tests - Romania, 4

Tags: number theory , diophantine equation

Solve in nonnegative integers the following equation : $$21^x+4^y=z^2$$

PEN H Problems, 19

Tags: diophantine equation

Find all $(x, y, z, n) \in {\mathbb{N}}^4$ such that $ x^3 +y^3 +z^3 =nx^2 y^2 z^2$.

1986 ITAMO, 6

Tags: diophantine , diophantine equation , number theory

Show that for any positive integer $n$ there exists an integer $m > 1$ such that $(\sqrt2-1)^n=\sqrt{m}-\sqrt{m-1}$.

PEN H Problems, 66

Tags: diophantine equation , inequalities

Let $b$ be a positive integer. Determine all $2002$-tuples of non-negative integers $(a_{1}, a_{2}, \cdots, a_{2002})$ satisfying \[\sum^{2002}_{j=1}{a_{j}}^{a_{j}}=2002{b}^{b}.\]

1998 Croatia National Olympiad, Problem 2

Tags: number theory , diophantine equation

Find all positive integer solutions of the equation $10(m+n)=mn$.

2009 QEDMO 6th, 1

Tags: diophantine , diophantine equation , number theory

Solve $y^5 - x^2 = 4$ in integers numbers $x,y$.

2022 Korea Junior Math Olympiad, 8

Tags: number theory , rational number , diophantine equation , pythagorean triple , discriminant

Find all pairs $(x, y)$ of rational numbers such that $$xy^2=x^2+2x-3$$

2004 Junior Balkan Team Selection Tests - Romania, 3

Tags: number theory , diophantine equation , diophantine

Let $p, q, r$ be primes and let $n$ be a positive integer such that $p^n + q^n = r^2$. Prove that $n = 1$. Laurentiu Panaitopol

2005 Swedish Mathematical Competition, 1

Tags: diophantine equation , number theory

Find all integer solutions $x$,$y$ of the equation $(x+y^2)(x^2+y)=(x+y)^3$.

2024 Polish MO Finals, 3

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

Determine all pairs $(p,q)$ of prime numbers with the following property: There are positive integers $a,b,c$ satisfying \[\frac{p}{a}+\frac{p}{b}+\frac{p}{c}=1 \quad \text{and} \quad \frac{a}{p}+\frac{b}{p}+\frac{c}{p}=q+1.\]

1973 Chisinau City MO, 70

Tags: diophantine , diophantine equation , number theory

The natural numbers $p, q$ satisfy the relation $p^p + q^q = p^q + q^p$. Prove that $p = q$.

2015 China Northern MO, 1

Tags: diophantine equation , diophantine , number theory

Find all integer solutions to the equation $$\frac{xyz}{w}+\frac{yzw}{x}+\frac{zwx}{y}+\frac{wxy}{z}=4$$

PEN H Problems, 76

Tags: diophantine equation

Find all pairs $(m,n)$ of integers that satisfy the equation \[(m-n)^{2}=\frac{4mn}{m+n-1}.\]

2009 Croatia Team Selection Test, 4

Tags: quadratic , diophantine equation , number theory

Prove that there are infinite many positive integers $ n$ such that $ n^2\plus{}1\mid n!$, and infinite many of those for which $ n^2\plus{}1 \nmid n!$.

2017 Regional Olympiad of Mexico Southeast, 4

Tags: number theory , diophantine equation , factorial

Find all couples of positive integers $m$ and $n$ such that $$n!+5=m^3$$

2011 NZMOC Camp Selection Problems, 6

Tags: diophantine equation , diophantine , number theory

Find all pairs of non-negative integers $m$ and $n$ that satisfy $$3 \cdot 2^m + 1 = n^2.$$

2014 CHKMO, 3

Tags: number theory , diophantine equation

Find all pairs $(a,b)$ of integers $a$ and $b$ satisfying \[(b^2+11(a-b))^2=a^3 b\]

1982 Tournament Of Towns, (025) 3

Tags: number theory , diophantine , diophantine equation , factorial

Prove that the equation $m!n! = k!$ has infinitely many solutions in which $m, n$ and $k$ are natural numbers greater than unity .

2023 Brazil National Olympiad, 4

Tags: number theory , diophantine equation , minimum value , algebra

Determine the smallest integer $k$ for which there are three distinct positive integers $a$, $b$ and $c$, such that $$a^2 =bc \text{ and } k = 2b+3c-a.$$

PEN H Problems, 64

Tags: diophantine equation

Show that there is no positive integer $k$ for which the equation \[(n-1)!+1=n^{k}\] is true when $n$ is greater than $5$.

PEN H Problems, 42

Tags: diophantine equation

Find all integers $a$ for which $x^3 -x+a$ has three integer roots.