Olympiad problems

If $p$ is a prime number and $x, y$ are positive integers, find in terms of $p$, all pairs $(x, y)$ that satisfy the equation: $$p(x -2) = x(y -1).$$ If $x+y = 21$, find all triples $(x, y, p)$ that satisfy this equation.

2010 Junior Balkan Team Selection Tests - Romania, 1

Tags: number theory , diophantine equation , diophantine

Determine the prime numbers $p, q, r$ with the property that: $p(p-7) + q (q-7) = r (r-7)$.

2020 Regional Competition For Advanced Students, 4

Tags: diophantine , diophantine equation , number theory

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)

1990 Poland - Second Round, 1

Tags: diophantine , number theory , diophantine equation

Find all pairs of integers $ x $, $ y $ satisfying the equation $$ (xy-1)^2 = (x +1)^2 + (y +1)^2.$$

2002 Singapore Senior Math Olympiad, 3

Tags: number theory , diophantine , radical , diophantine equation

Prove that for natural numbers $p$ and $q$, there exists a natural number $x$ such that $$(\sqrt{p}+\sqrt{p-1})^q=\sqrt{x}+\sqrt{x-1}$$ (As an example, if $p = 3, q = 2$, then $x$ can be taken to be $25$.)

2022 New Zealand MO, 1

Tags: number theory , diophantine equation , diophantine

Find all integers $a, b$ such that $$a^2 + b = b^{2022}.$$

2013 Czech-Polish-Slovak Junior Match, 1

Tags: number theory , diophantine equation , diophantine , radical

Determine all pairs $(x, y)$ of integers for which satisfy the equality $\sqrt{x-\sqrt{y}}+ \sqrt{x+\sqrt{y}}= \sqrt{xy}$

2009 Postal Coaching, 2

Tags: diophantine , diophantine equation , number theory

Determine, with proof, all the integer solutions of the equation $x^3 + 2y^3 + 4z^3 - 6xyz = 1$.

Mathley 2014-15, 7

Tags: diophantine , prime , number theory , diophantine equation

Find all primes $p,q, r$ such that $\frac{p^{2q}+q^{2p}}{p^3-pq+q^3} = r$. Titu Andreescu, Mathematics Department, College of Texas, USA

2011 Cuba MO, 2

Tags: number theory , diophantine , diophantine equation

Determine all the integer solutions of the equation $3x^4-2024y+1= 0$.

2017 Rioplatense Mathematical Olympiad, Level 3, 1

Tags: number theory , diophantine , diophantine equation

Let $a$ be a fixed positive integer. Find the largest integer $b$ such that $(x+a)(x+b)=x+a+b$, for some integer $x$.

2011 Swedish Mathematical Competition, 1

Tags: diophantine , number theory , diophantine equation

Determine all positive integers $k$, $\ell$, $m$ and $n$, such that $$\frac{1}{k!}+\frac{1}{\ell!}+\frac{1}{m!} =\frac{1}{n!} $$

VMEO III 2006, 10.2

Tags: diophantine , number theory , diophantine equation

Find all triples of integers $(x, y, z)$ such that $x^4 + 5y^4 = z^4$.

2005 Thailand Mathematical Olympiad, 6

Tags: diophantine , number theory , diophantine equation

Find the number of positive integer solutions to the equation $(x_1+x_2+x_3)^2(y_1+y_2) = 2548$.

1980 Polish MO Finals, 2

Tags: number theory , diophantine , diophantine equation

Prove that for every $n$ there exists a solution of the equation $$a^2 +b^2 +c^2 = 3abc$$ in natural numbers $a,b,c$ greater than $n$.

1966 Poland - Second Round, 1

Tags: number theory , diophantine

Solve the equation in natural numbers $$ x+y+z+t=xyzt. $$