NMC 2006

The real numbers $x, y$ and $z$ are not all equal and satisfy

$$x+\frac{1}{y}=y+\frac{1}{z}=z+\frac{1}{x}=k$$

Determine all possible values of $k$.

A sequence of positive integers $\left\{a_{n}\right\}$ is given by

$$a_{0}=m \text { and } a_{n+1}=a_{n}^{5}+487$$

for all $n \geq 0$. Determine all values of $m$ for which the sequence contains as many square numbers as possible.

The squares of a $100 \times 100$ chessboard are painted with 100 different colours. Each square has only one colour and every colour is used exactly 100 times. Show that there exists a row or a column on the chessboard in which at least 10 colours are used.