Olympiad problems

2007 Argentina National Olympiad, 1

Find all the prime numbers $p$ and $q$ such that $ p^2+q=37q^2+p $. Clarification: $1$ is not a prime number.

1997 Estonia National Olympiad, 1

Tags: diophantine equation , diophantine , number theory

Prove that for every integer $n\ge 3$ there are such positives integers $x$ and $y$ such that $2^n = 7x^2 + y^2$

2017 Istmo Centroamericano MO, 4

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

Suppose that $a$ and $ b$ are distinct positive integers satisfying $20a + 17b = p$ and $17a + 20b = q$ for certain primes $p$ and $ q$. Determine the minimum value of $p + q$.

1998 Switzerland Team Selection Test, 2

Tags: diophantine , diophantine equation , number theory

Find all nonnegative integer solutions $(x,y,z)$ of the equation $\frac{1}{x+2}+\frac{1}{y+2}=\frac{1}{2} +\frac{1}{z+2}$

1942 Eotvos Mathematical Competition, 2

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

Let $a, b, c $and $d$ be integers such that for all integers m and n, there exist integers $x$ and $y$ such that $ax + by = m$, and $cx + dy = n$. Prove that $ad - bc = \pm 1$.

2005 Austria Beginners' Competition, 1

Tags: number theory , diophantine equation , diophantine

Show that there are no positive integers $a$ und $b$ such that $4a(a + 1) = b(b + 3)$

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), 2b

Tags: diophantine , diophantine equation , number theory

Find all integers $x,y,z$ such that $x^3 +5y^3 = 9z^3$.

VI Soros Olympiad 1999 - 2000 (Russia), 9.2

Tags: diophantine equation , diophantine , number theory

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

1981 All Soviet Union Mathematical Olympiad, 316

Tags: number theory , diophantine equation , diophantine

Find the natural solutions of the equation $x^3 - y^3 = xy + 61$.

2020 Abels Math Contest (Norwegian MO) Final, 2a

Tags: diophantine , diophantine equation , number theory

Find all natural numbers $k$ such that there exist natural numbers $a_1,a_2,...,a_{k+1}$ with $ a_1!+a_2!+... +a_{k+1}!=k!$ Note that we do not consider $0$ to be a natural number.

2011 Saudi Arabia Pre-TST, 3.2

Tags: diophantine , number theory , diophantine equation

Find all pairs of nonnegative integers $(a, b)$ such that $a+2b-b^2=\sqrt{2a+a^2+|2a+1-2b|}$.

1992 Rioplatense Mathematical Olympiad, Level 3, 2

Tags: diophantine equation , diophantine , number theory

Determine the integers $0 \le a \le b \le c \le d$ such that: $$2^n= a^2 + b^2 + c^2 + d^2.$$

1978 Dutch Mathematical Olympiad, 1

Tags: diophantine equation , diophantine , number theory

Prove that no integer $x$ and $y$ satisfy: $$3x^2 = 9 + y^3.$$

2018 Ecuador NMO (OMEC), 1

Tags: diophantine , number theory , diophantine equation

Let $a, b$ be integers. Show that the equation $a^2 + b^2 = 26a$ has at least $12$ solutions.

1971 Dutch Mathematical Olympiad, 3

Tags: number theory , diophantine equation , diophantine

Prove that $(0,1)$, $(0, -1)$,$( -1,1)$ and $(-1,-1)$ are the only integer solutions of $$x^2 + x +1 = y^2.$$

2015 Saudi Arabia JBMO TST, 1

Tags: number theory , diophantine equation , diophantine

Find all the triples $(x,y,z)$ of positive integers such that $xy+yz+zx-xyz=2015$

VI Soros Olympiad 1999 - 2000 (Russia), 11.4

Tags: diophantine , diophantine equation , number theory

For prime numbers $p$ and $q$, natural numbers $n$, $k$, $r$, the equality $p^{2k}+q^{2n}=r^2$ holds. Prove that the number $r$ is prime.

2015 Costa Rica - Final Round, 4

Tags: number theory , diophantine equation , diophantine

Find all triples $(p,M, z)$ of integers, where $p$ is prime, $m$ is positive and $z$ is negative, that satisfy the equation $$p^3 + pm + 2zm = m^2 + pz + z^2$$

2016 Balkan MO Shortlist, N3

Tags: diophantine equation , number theory , diophantine

Find all the integer solutions $(x,y,z)$ of the equation $(x + y + z)^5 = 80xyz(x^2 + y^2 + z^2)$,

2014 Junior Balkan Team Selection Tests - Moldova, 2

Tags: diophantine , diophantine equation , number theory

Determine all pairs of integers $(x, y)$ that satisfy equation $(y - 2) x^2 + (y^2 - 6y + 8) x = y^2 - 5y + 62$.

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.

2006 Abels Math Contest (Norwegian MO), 3

Tags: rational , number theory , diophantine , integer

(a) Let $a$ and $b$ be rational numbers such that line $y = ax + b$ intersects the circle $x^2 + y^2 = 5$ at two different points. Show that if one of the intersections has two rational coordinates, so does the other intersection. (b) Show that there are infinitely many triples ($k, n, m$) that are such that $k^2 + n^2 = 5m^2$, where $k, n$ and $m$ are integers, and not all three have any in common prime factor.

2011 Hanoi Open Mathematics Competitions, 6

Tags: diophantine , diophantine equation , number theory

Find all positive integers $(m,n)$ such that $m^2 + n^2 + 3 = 4(m + n)$

1999 Austrian-Polish Competition, 7

Tags: diophantine equation , diophantine , number theory

Find all pairs $(x,y)$ of positive integers such that $x^{x+y} =y^{y-x}$.