Olympiad problems

2000 Swedish Mathematical Competition, 3

Are there any integral solutions to $n^2 + (n+1)^2 + (n+2)^2 = m^2$ ?

2003 Junior Balkan Team Selection Tests - Romania, 3

Tags: diophantine , number theory , radical , inequalities

Let $n$ be a positive integer. Prove that there are no positive integers $x$ and $y$ such as $\sqrt{n}+\sqrt{n+1} < \sqrt{x}+\sqrt{y} <\sqrt{4n+2} $

2013 Estonia Team Selection Test, 1

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

Find all prime numbers $p$ for which one can find a positive integer $m$ and nonnegative integers $a_0,a_1,...,a_m$ less than $p$ such that $$\begin{cases} a_0+a_1p+...+a_{m-1}p^{m-1}+a_{m}p^{m} = 2013 \\ a_0+a_1+...+a_{m-1}+a_{m} = 11\end{cases}$$

2003 Abels Math Contest (Norwegian MO), 2a

Tags: diophantine equation , diophantine , number theory

Find all pairs $(x, y)$ of integers numbers such that $y^3+5=x(y^2+2)$

1992 Swedish Mathematical Competition, 4

Tags: inequalities , number theory , diophantine

Find all positive integers $a, b, c$ such that $a < b$, $a < 4c$, and $b c^3 \le a c^3 + b$.

2010 Saudi Arabia BMO TST, 4

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

Find all triples $(x,y, z)$ of integers such that $$\begin{cases} x^2y + y^2z + z^2x= 2010^2 \\ xy^2 + yz^2 + zx^2= -2010 \end{cases}$$

2007 Swedish Mathematical Competition, 1

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

Solve the following system \[ \left\{ \begin{array}{l} xyzu-x^3=9 \\ x+yz=\dfrac{3}{2}u \\ \end{array} \right. \] in positive integers $x$, $y$, $z$ and $u$.

VI Soros Olympiad 1999 - 2000 (Russia), 9.4

Tags: number theory , diophantine equation , diophantine

Are there integers $k$ and $m$ for which $$\frac{(k-3)(k-2)(k-1)k+1}{(k+1)(k+2)(k+3)(k+4)+1}=m(m+1)+(m+1)(m+2)+(m+2)m \,\, ?$$

2005 Estonia Team Selection Test, 3

Tags: diophantine equation , diophantine , number theory

Find all pairs $(x, y)$ of positive integers satisfying the equation $(x + y)^x = x^y$.

2014 Estonia Team Selection Test, 6

Tags: diophantine , diophantine equation , number theory

Find all natural numbers $n$ such that the equation $x^2 + y^2 + z^2 = nxyz$ has solutions in positive integers

1999 German National Olympiad, 1

Tags: diophantine equation , diophantine , number theory

Find all $x,y$ which satisfy the equality $x^2 +xy+y^2 = 97$, when $x,y$ are a) natural numbers, b) integers

2019 Romania National Olympiad, 3

Tags: algebra , diophantine , equation

Prove that the number of solutions in $ \left( \mathbb{N}\cup\{ 0 \} \right)\times \left( \mathbb{N}\cup\{ 0 \} \right)\times \left( \mathbb{N}\cup\{ 0 \} \right) $ of the parametric equation $$ \sqrt{x^2+y+n}+\sqrt{y^2+x+n} = z, $$ is greater than zero and finite, for nay natural number $ n. $

2010 Junior Balkan Team Selection Tests - Romania, 1

Tags: diophantine equation , diophantine , number theory

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

2017 Hanoi Open Mathematics Competitions, 3

Tags: diophantine equation , diophantine , number theory

The number of real triples $(x , y , z )$ that satisfy the equation $x^4 + 4y^4 + z^4 + 4 = 8xyz$ is (A): $0$, (B): $1$, (C): $2$, (D): $8$, (E): None of the above.

2013 Estonia Team Selection Test, 1

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

Find all prime numbers $p$ for which one can find a positive integer $m$ and nonnegative integers $a_0,a_1,...,a_m$ less than $p$ such that $$\begin{cases} a_0+a_1p+...+a_{m-1}p^{m-1}+a_{m}p^{m} = 2013 \\ a_0+a_1+...+a_{m-1}+a_{m} = 11\end{cases}$$