Olympiad problems

2001 Junior Balkan Team Selection Tests - Moldova, 5

Determine if there is a non-natural natural number $n$ with the property that $\sqrt{n + 1} + \sqrt{n - 1}$ is rational.

2017 Czech-Polish-Slovak Junior Match, 6

Tags: integer , rational , number theory

On the board are written $100$ mutually different positive real numbers, such that for any three different numbers $a, b, c$ is $a^2 + bc$ is an integer. Prove that for any two numbers $x, y$ from the board , number $\frac{x}{y}$ is rational.

VII Soros Olympiad 2000 - 01, 10.2

Tags: trigonometry , rational , algebra

Let $a$ and $ b$ be acute corners. Prove that if $\sin a$, $\sin b$, and $\sin (a + b)$ are rational numbers, then $\cos a$, $\cos b$, and $\cos (a + b)$ are also rational numbers.

2010 NZMOC Camp Selection Problems, 5

Tags: number theory , algebra , rational

Determine the values of the positive integer $n$ for which $$A =\sqrt{\frac{9n - 1}{n + 7}}$$ is rational.

2007 Germany Team Selection Test, 3

Tags: number theory , rational , decimal representation

For $ x \in (0, 1)$ let $ y \in (0, 1)$ be the number whose $ n$-th digit after the decimal point is the $ 2^{n}$-th digit after the decimal point of $ x$. Show that if $ x$ is rational then so is $ y$. [i]Proposed by J.P. Grossman, Canada[/i]

2001 Swedish Mathematical Competition, 2

Tags: algebra , rational , radical

Show that $\sqrt[3]{\sqrt{52} + 5}- \sqrt[3]{\sqrt{52}- 5}$ is rational.