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{\large \bf 36th Canadian Mathematical Olympiad}\\

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Wednesday, March 31, 2004\\
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\begin{enumerate}
\item Find all ordered triples $(x,y,z)$ of real numbers which satisfy
 the following system of equations:
 \[ \left\{ \begin{array}{rcl}
  xy & = & z-x-y \\ xz & = & y-x-z \\ yz & = & x-y-z
 \end{array} \right. \]

\item \parbox[t]{4.5in}{How many ways can 8 mutually non-attacking rooks be placed
 on the $9\times9$ 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.]}
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\item Let $A,B,C,D$ be four points on a circle (occurring in clockwise order),
 with $AB<AD$ and $BC>CD$.
Let the bisector of angle $BAD$ meet the circle at $X$ and the bisector of angle $BCD$
 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 $BD$ must be a diameter of the circle.

\item Let $p$ be an odd prime.
 Prove that $$\sum_{k = 1}^{p - 1} k^{2p - 1} \equiv \frac{p(p + 1)}{2} \pmod{p^2}.$$

 [Note that $a\equiv b\!\!\pmod{m}$ means that $a-b$ is divisible by $m$.]

\item 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 $S$?
\end{enumerate}

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