CMO 2004

Find all ordered triples $(x, y, z)$ of real numbers which satisfy the following system of equations:

$$\left\{\begin{array}{l} x y=z-x-y \\ x z=y-x-z \\ y z=x-y-z \end{array}\right.$$

How many ways can 8 mutually non-attacking rooks be placed on the $9 \times 9$ chessboard (shown here) so that all 8 rooks are on squares of the same colour?

[Two rooks are said to be attacking each other if they are placed in the same row or column of the board.]

Let $A, B, C, D$ be four points on a circle (occurring in clockwise order), with $A B<A D$ and $B C>C D$. Let the bisector of angle $B A D$ meet the circle at $X$ and the bisector of angle $B C D$ meet the circle at $Y$. Consider the hexagon formed by these six points on the circle. If four of the six sides of the hexagon have equal length, prove that $B D$ must be a diameter of the circle.

Let $p$ be an odd prime. Prove that

$$\sum_{k=1}^{p-1} k^{2 p-1} \equiv \frac{p(p+1)}{2} \left(\bmod p^{2}\right)$$

[Note that $a \equiv b(\bmod m)$ means that $a-b$ is divisible by $m$.]

Let $T$ be the set of all positive integer divisors of $2004^{100}$. What is the largest possible number of elements that a subset $S$ of $T$ can have if no element of $S$ is an integer multiple of any other element of $S$ ?