Olympiad problems

2024 Ecuador NMO (OMEC), 5

Find all triples of non-negative integer numbers $(E, C, U)$ such that $EC \ge 1$ and: $$2^{3^E}+3^{2^C}=593 \cdot 5^U$$

1998 Slovenia National Olympiad, Problem 1

Tags: number theory , diophantine equation

Find all integers $x,y$ which satisfy the equation $xy=20-3x+y$.

1984 All Soviet Union Mathematical Olympiad, 379

Tags: diophantine , diophantine equation , number theory , algebra , exponential

Find integers $m$ and $n$ such that $(5 + 3 \sqrt2)^m = (3 + 5 \sqrt2)^n$.

PEN H Problems, 36

Tags: diophantine equation

Prove that the equation $a^2 +b^2 =c^2 +3$ has infinitely many integer solutions $(a, b, c)$.

2014 Ukraine Team Selection Test, 9

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

Let $m, n$ be odd prime numbers. Find all pairs of integers numbers $a, b$ for which the system of equations: $x^m+y^m+z^m=a$, $x^n+y^n+z^n=b$ has many solutions in integers $x, y, z$.

1997 Brazil Team Selection Test, Problem 2

Tags: diophantine equation , number theory

We say that a subset $A$ of $\mathbb N$ is good if for some positive integer $n$, the equation $x-y=n$ admits infinitely many solutions with $x,y\in A$. If $A_1,A_2,\ldots,A_{100}$ are sets whose union is $\mathbb N$, prove that at least one of the $A_i$s is good.

2017 Harvard-MIT Mathematics Tournament, 5

Tags: number theory , diophantine equation

Find the number of ordered triples of positive integers $(a, b, c)$ such that \[6a + 10b + 15c = 3000.\]

2019 Serbia Team Selection Test, P5

Tags: diophantine equation , number theory

Solve the equation in nonnegative integers:\\ $2^x=5^y+3$

PEN H Problems, 27

Tags: diophantine equation , perimeter , inradius , modular arithmetic , geometry

Prove that there exist infinitely many positive integers $n$ such that $p=nr$, where $p$ and $r$ are respectively the semi-perimeter and the inradius of a triangle with integer side lengths.

2004 Argentina National Olympiad, 2

Tags: diophantine equation , number theory , system of equations

Determine all positive integers $a,b,c,d$ such that$$\begin{cases} a<b \\ a^2c =b^2d \\ ab+cd =2^{99}+2^{101} \end{cases}$$

2014 VTRMC, Problem 5

Tags: diophantine equation , number theory

Let $n\ge1$ and $r\ge2$ be positive integers. Prove that there is no integer $m$ such that $n(n+1)(n+2)=m^r$.

2014 China Northern MO, 3

Tags: diophantine equation , number theory

Determine whether there exist an infinite number of positive integers $x,y $ satisfying the condition: $x^2+y \mid x+y^2.$ Please prove it.

2016 NZMOC Camp Selection Problems, 7

Tags: diophantine , diophantine equation , number theory

Find all positive integers $n$ for which the equation $$(x^2 + y^2)^n = (xy)^{2016}$$ has positive integer solutions.

PEN H Problems, 84

Tags: diophantine equation

For what positive numbers $a$ is \[\sqrt[3]{2+\sqrt{a}}+\sqrt[3]{2-\sqrt{a}}\] an integer?

2017 Singapore Senior Math Olympiad, 1

Tags: diophantine , diophantine equation , number theory , factorial

Let $n$ be a positive integer and $a_1,a_2,...,a_{2n}$ be $2n$ distinct integers. Given that the equation $|x-a_1| |x-a_2| ... |x-a_{2n}| =(n!)^2$ has an integer solution $x = m$, find $m$ in terms of $a_1,a_2,...,a_{2n}$

1980 IMO Longlists, 3

Tags: number theory , equation , algebra , diophantine equation

Prove that the equation \[ x^n + 1 = y^{n+1}, \] where $n$ is a positive integer not smaller then 2, has no positive integer solutions in $x$ and $y$ for which $x$ and $n+1$ are relatively prime.

PEN H Problems, 7

Tags: diophantine equation

Determine all pairs $(x,y)$ of positive integers satisfying the equation \[(x+y)^{2}-2(xy)^{2}=1.\]

2018 Hanoi Open Mathematics Competitions, 11

Tags: diophantine equation , number theory

Find all pairs of nonnegative integers $(x, y)$ for which $(xy + 2)^2 = x^2 + y^2 $.

2018 Finnish National High School Mathematics Comp, 5

Tags: number theory , diophantine equation , diophantine

Solve the diophantine equation $x^{2018}-y^{2018}=(xy)^{2017}$ when $x$ and $y$ are non-negative integers.

2007 Indonesia MO, 2

Tags: relatively prime , diophantine equation , number theory

For every positive integer $ n$, $ b(n)$ denote the number of positive divisors of $ n$ and $ p(n)$ denote the sum of all positive divisors of $ n$. For example, $ b(14)\equal{}4$ and $ p(14)\equal{}24$. Let $ k$ be a positive integer greater than $ 1$. (a) Prove that there are infinitely many positive integers $ n$ which satisfy $ b(n)\equal{}k^2\minus{}k\plus{}1$. (b) Prove that there are finitely many positive integers $ n$ which satisfy $ p(n)\equal{}k^2\minus{}k\plus{}1$.

1984 Swedish Mathematical Competition, 5

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

Solve in natural numbers $a,b,c$ the system \[\left\{ \begin{array}{l}a^3 -b^3 -c^3 = 3abc \\ a^2 = 2(a+b+c)\\ \end{array} \right. \]

2014 Belarusian National Olympiad, 2

Tags: number theory , diophantine , diophantine equation

Pairwise distinct prime numbers $p, q, r$ satisfy the equality $$rp^3 + p^2 + p = 2rq^2 +q^2 + q.$$ Determine all possible values of the product $pqr$.

1987 Poland - Second Round, 5

Tags: number theory , diophantine , diophantine equation

Determine all prime numbers $ p $ and natural numbers $ x, y $ for which $ p^x-y^3 = 1 $.

1989 IberoAmerican, 3

Tags: modular arithmetic , diophantine equation , number theory

Show that the equation $2x^2-3x=3y^2$ has infinitely many solutions in positive integers.

1986 IMO Longlists, 18

Tags: diophantine equation , equation , algebra , number theory

Provided the equation $xyz = p^n(x + y + z)$ where $p \geq 3$ is a prime and $n \in \mathbb{N}$. Prove that the equation has at least $3n + 3$ different solutions $(x,y,z)$ with natural numbers $x,y,z$ and $x < y < z$. Prove the same for $p > 3$ being an odd integer.