NL-NWO 2011

Determine all triples of positive integers $(a, b, n)$ that satisfy the following equation: $a! + b! = 2^n$

Let $ABC$ be a triangle. Points $P$ and $Q$ lie on side $BC$ and satisfy $|BP| =|PQ| = |QC| = \frac13 |BC|$ . Points $R$ and $S$ lie on side $CA$ and satisfy $|CR| =|RS| = |SA| = 1 3 |CA|$ . Finally, points $T$ and $U$ lie on side $AB$ and satisfy $|AT| = |TU| = |UB| =\frac13 |AB|$ . Points $P, Q,R, S, T$ and $U$ turn out to lie on a common circle. Prove that $ABC$ is an equilateral triangle.

In a tournament among six teams, every team plays against each other team exactly once. When a team wins, it receives $3$ points and the losing team receives $0$ points. If the game is a draw, the two teams receive $1$ point each. Can the final scores of the six teams be six consecutive numbers $a,a +1,...,a + 5$ ? If so, determine all values of $a$ for which this is possible.

Determine all pairs of positive real numbers $(a, b)$ with $a > b$ that satisfy the following equations: $a\sqrt{a}+ b\sqrt{b} = 134$ and $a\sqrt{b}+ b\sqrt{a} = 126$ .

The number devil has coloured the integer numbers: every integer is coloured either black or white. The number $1$ is coloured white. For every two white numbers $a$ and $b$ ( $a$ and $b$ are allowed to be equal) the numbers $a-b$ and $a +$ b have different colours. Prove that $2011$ is coloured white.