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{\large \bf The Canadian Mathematical Olympiad - 2003}\\
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%Wednesday, March 26\\
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\begin{enumerate}
\item Consider a standard twelve-hour clock whose hour and minute hands move continuously.
Let $m$ be an integer, with $1 \leq m \leq 720$.
At precisely $m$ minutes after 12:00, the angle made by the hour hand and
minute hand is exactly $1^\circ$.
Determine all possible values of $m$.

\item Find the last three digits of the number $2003^{{2002}^{2001}}$.

\item Find all real positive solutions (if any) to
 \[ x^3+y^3+z^3 = x+y+z, \mbox{ and} \]
 \[ x^2+y^2+z^2 = xyz. \]

\item Prove that when three circles share the same chord $AB$,
 every line through $A$ different from $AB$ determines the same ratio $XY\!: \,YZ$,
 where $X$ is an arbitrary point different from $B$ on the first circle
 while $Y$ and $Z$ are the points where $AX$ intersects the other two circles
 (labelled so that $Y$ is between $X$ and $Z$).
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\item Let $S$ be a set of $n$ points in the plane such that any two points of
$S$ are at least $1$ unit apart.
Prove there is a subset $T$ of $S$ with at least $n/7$
%$\displaystyle {n\over7}$
points such that any two points of $T$ are at least $\sqrt{3}$ units apart.

\end{enumerate}

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