Olympiad problems

2017 Harvard-MIT Mathematics Tournament, 4

Find the number of ordered triples of nonnegative integers $(a, b, c)$ that satisfy \[(ab + 1)(bc + 1)(ca + 1) = 84.\]

VI Soros Olympiad 1999 - 2000 (Russia), 9.2

Tags: number theory , diophantine equation , diophantine

Find the smallest natural number n such that for all integers $m > n$ there are positive integers $x$ and $y$ for which the equality 1$7x + 23y = m$ holds

2005 Austrian-Polish Competition, 4

Tags: perfect square , fermat's little theorem , prime , number theory , diophantine equation

Determine the smallest natural number $a\geq 2$ for which there exists a prime number $p$ and a natural number $b\geq 2$ such that \[\frac{a^p - a}{p}=b^2.\]

1999 Singapore Team Selection Test, 1

Tags: diophantine equation , diophantine , number theory

Find all integers $m$ for which the equation $$x^3 - mx^2 + mx - (m^2 + 1) = 0$$ has an integer solution.

PEN H Problems, 44

Tags: diophantine equation , calculus , integration

For all $n \in \mathbb{N}$, show that the number of integral solutions $(x, y)$ of \[x^{2}+xy+y^{2}=n\] is finite and a multiple of $6$.

2011 India IMO Training Camp, 1

Tags: 3d geometry , diophantine equation , number theory , geometry

Find all positive integer $n$ satisfying the conditions $a)n^2=(a+1)^3-a^3$ $b)2n+119$ is a perfect square.

2018 Hanoi Open Mathematics Competitions, 6

Tags: algebra , diophantine , diophantine equation

Nam spent $20$ dollars for $20$ stationery items consisting of books, pens and pencils. Each book, pen, and pencil cost $3$ dollars, $1.5$ dollars and $0.5$ dollar respectively. How many dollars did Nam spend for books?

2018 Bosnia and Herzegovina Junior BMO TST, 2

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

Find all integer triples $(p,m,n)$ that satisfy: $p^m-n^3=27$ where $p$ is a prime number.

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: \[a^p+b^p=p^c.\]

1997 Slovenia National Olympiad, Problem 1

Tags: diophantine equation , number theory

Marko chose two prime numbers $a$ and $b$ with the same number of digits and wrote them down one after another, thus obtaining a number $c$. When he decreased $c$ by the product of $a$ and $b$, he got the result $154$. Determine the number $c$.

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$.

2002 Junior Balkan Team Selection Tests - Romania, 2

Tags: number theory , diophantine , diophantine equation , divisor

Let $k,n,p$ be positive integers such that $p$ is a prime number, $k < 1000$ and $\sqrt{k} = n\sqrt{p}$. a) Prove that if the equation $\sqrt{k + 100x} = (n + x)\sqrt{p}$ has a non-zero integer solution, then $p$ is a divisor of $10$. b) Find the number of all non-negative solutions of the above equation.

1996 Estonia National Olympiad, 1

Tags: prime , diophantine equation , diophantine , number theory

Let $p$ be a fixed prime. Find all pairs $(x,y)$ of positive numbers satisfying $p(x-y) = xy$.

1994 Abels Math Contest (Norwegian MO), 2a

Tags: diophantine , number theory , diophantine equation , prime

Find all primes $p,q,r$ and natural numbers $n$ such that $\frac{1}{p}+\frac{1}{q}+\frac{1}{r}=\frac{1}{n}$.

1988 All Soviet Union Mathematical Olympiad, 471

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

Find all positive integers $n$ satisfying $\left(1 +\frac{1}{n}\right)^{n+1} = \left(1 + \frac{1}{1998}\right)^{1998}$.

PEN H Problems, 18

Tags: diophantine equation

Determine all positive integer solutions $(x, y, z, t)$ of the equation \[(x+y)(y+z)(z+x)=xyzt\] for which $\gcd(x, y)=\gcd(y, z)=\gcd(z, x)=1$.

1989 IMO Shortlist, 31

Tags: algebra , diophantine equation , linear equation , counting

Let $ a_1 \geq a_2 \geq a_3 \in \mathbb{Z}^\plus{}$ be given and let N$ (a_1, a_2, a_3)$ be the number of solutions $ (x_1, x_2, x_3)$ of the equation \[ \sum^3_{k\equal{}1} \frac{a_k}{x_k} \equal{} 1.\] where $ x_1, x_2,$ and $ x_3$ are positive integers. Prove that \[ N(a_1, a_2, a_3) \leq 6 a_1 a_2 (3 \plus{} ln(2 a_1)).\]

2012 Dutch BxMO/EGMO TST, 3

Tags: number theory , diophantine equation

Find all pairs of positive integers $(x,y) $ for which $x^3 + y^3 = 4(x^2y + xy^2 - 5) .$

2015 Middle European Mathematical Olympiad, 7

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

Find all pairs of positive integers $(a,b)$ such that $$a!+b!=a^b + b^a.$$

2022 Azerbaijan National Mathematical Olympiad, 3

Tags: diophantine equation , diophantine , number theory

Let $A$ be the set of all triples $(x, y, z)$ of positive integers satisfying $2x^2 + 3y^3 = 4z^4$ . a) Show that if $(x, y, z) \in A$ then $6$ divides all of $x, y, z$. b) Show that $A$ is an infinite set.

2017 OMMock - Mexico National Olympiad Mock Exam, 4

Tags: number theory , diophantine equation

Show that the equation $$a^2b=2017(a+b)$$ has no solutions for positive integers $a$ and $b$. [i]Proposed by Oriol Solé[/i]

2015 USAMO, 1

Tags: diophantine equation , number theory , algebra

Solve in integers the equation \[ x^2+xy+y^2 = \left(\frac{x+y}{3}+1\right)^3. \]

1986 IMO Longlists, 76

Tags: number theory , diophantine equation , geometry , pythagorean theorem , algebra

Let $A, B$, and $C$ be three points on the edge of a circular chord such that $B$ is due west of $C$ and $ABC$ is an equilateral triangle whose side is $86$ meters long. A boy swam from $A$ directly toward $B$. After covering a distance of $x$ meters, he turned and swam westward, reaching the shore after covering a distance of $y$ meters. If $x$ and $y$ are both positive integers, determine $y.$

1995 ITAMO, 6

Tags: diophantine equation , diophantine , number theory

Find all pairs of positive integers $x,y$ such that $x^2 +615 = 2^y$

2005 Austrian-Polish Competition, 9

Tags: number theory , diophantine equation , sum of cubes , infinitely many solutions

Consider the equation $x^3 + y^3 + z^3 = 2$. a) Prove that it has infinitely many integer solutions $x,y,z$. b) Determine all integer solutions $x, y, z$ with $|x|, |y|, |z| \leq 28$.