Olympiad problems

2006 Hanoi Open Mathematics Competitions, 3

Find the number of different positive integer triples $(x, y,z)$ satisfying the equations $x^2 + y -z = 100$ and $x + y^2 - z = 124$:

2011 Saudi Arabia Pre-TST, 3.2

Tags: diophantine equation , diophantine , number theory

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

2015 India Regional MathematicaI Olympiad, 3

Tags: number theory , diophantine equation , system of equations

Find all integers $a,b,c$ such that $a^{2}=bc+4$ and $b^{2}=ca+4$.

2017 Kyiv Mathematical Festival, 5

Tags: diophantine equation , number theory

Find all the pairs of integers $(x,y)$ for which $(x^2+y)(y^2+x)=(x+1)(y+1).$

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 $$p^3 + pm + 2zm = m^2 + pz + z^2$$

2022 3rd Memorial "Aleksandar Blazhevski-Cane", P3

Tags: number theory , diophantine equation , 3 variables

Find all triplets of positive integers $(x, y, z)$ such that $x^2 + y^2 + x + y + z = xyz + 1$. [i]Proposed by Viktor Simjanoski[/i]

1998 Slovenia Team Selection Test, 4

Tags: gcd , number theory , diophantine equation , diophantine

Find all positive integers $x$ and $y$ such that $x+y^2+z^3 = xyz$, where $z$ is the greatest common divisor of $x$ and $y$

1996 Estonia National Olympiad, 1

Tags: sum , product , diophantine equation , diophantine , number theory

Find all pairs of integers $(x, y)$ such that ths sum of the fractions $\frac{19}{x}$ and $\frac{96}{y}$ would be equal to their product.

2013 JBMO Shortlist, 5

Tags: positive integer , number theory , diophantine equation

Solve in positive integers: $\frac{1}{x^2}+\frac{y}{xz}+\frac{1}{z^2}=\frac{1}{2013}$ .

2000 IMO Shortlist, 5

Tags: diophantine equation , number theory , inradius , perimeter , triangle

Prove that there exist infinitely many positive integers $ n$ such that $ p \equal{} nr,$ where $ p$ and $ r$ are respectively the semiperimeter and the inradius of a triangle with integer side lengths.

2016 Junior Balkan Team Selection Tests - Moldova, 6

Tags: diophantine , diophantine equation , number theory

Determine all pairs $(x, y)$ of natural numbers satisfying the equation $5^x=y^4+4y+1$.

2004 Singapore MO Open, 2

Tags: number theory , diophantine , diophantine equation

Find the number of ordered pairs $(a, b)$ of integers, where $1 \le a, b \le 2004$, such that $x^2 + ax + b = 167 y$ has integer solutions in $x$ and $y$. Justify your answer.

1988 IMO Shortlist, 9

Tags: number theory , divisibility , diophantine equation , perfect square , vieta jumping

Let $ a$ and $ b$ be two positive integers such that $ a \cdot b \plus{} 1$ divides $ a^{2} \plus{} b^{2}$. Show that $ \frac {a^{2} \plus{} b^{2}}{a \cdot b \plus{} 1}$ is a perfect square.

2002 Abels Math Contest (Norwegian MO), 1b

Tags: diophantine , diophantine equation , number theory

Find all integers $c$ such that the equation $(2a+b) (2b+a) =5^c$ has integer solutions.

2003 Chile National Olympiad, 2

Tags: number theory , diophantine , diophantine equation

Find all primes $p, q$ such that $p + q = (p-q)^3$.

2007 Indonesia TST, 3

Tags: inequalities , floor function , diophantine equation , number theory

For each real number $ x$< let $ \lfloor x \rfloor$ be the integer satisfying $ \lfloor x \rfloor \le x < \lfloor x \rfloor \plus{}1$ and let $ \{x\}\equal{}x\minus{}\lfloor x \rfloor$. Let $ c$ be a real number such that \[ \{n\sqrt{3}\}>\dfrac{c}{n\sqrt{3}}\] for all positive integers $ n$. Prove that $ c \le 1$.

KoMaL A Problems 2023/2024, A. 858

Tags: quadratic residue , number theory , diophantine equation

Prove that the only integer solution of the following system of equations is $u=v=x=y=z=0$: $$uv=x^2-5y^2, (u+v)(u+2v)=x^2-5z^2$$

1979 Brazil National Olympiad, 4

Tags: number theory , diophantine equation

Show that the number of positive integer solutions to $x_1 + 2^3x_2 + 3^3x_3 + \ldots + 10^3x_{10} = 3025$ (*) equals the number of non-negative integer solutions to the equation $y_1 + 2^3y_2 + 3^3y_3 + \ldots + 10^3y_{10} = 0$. Hence show that (*) has a unique solution in positive integers and find it.