\documentclass[12pt]{article}
\input epsf.sty
%\topmargin -.85 in
\oddsidemargin -0.3 in
\headheight 0 in
\parskip 0.3cm
\parindent 0 cm
\evensidemargin 0 in
\textwidth 7 in
\textheight 9.5 in
\thispagestyle{empty}
\begin{document}
\setlength{\unitlength}{1in}
\begin{center}
{\Large \bf THE 2002 CANADIAN MATHEMATICAL OLYMPIAD}
\end{center}
%\begin{picture}(8.5,1.75)
%\put(6.7,1.55){\shortstack{{\LARGE \bf 2002}}}
%\put(.7,0){\shortstack{
% {\LARGE \bf The 2002 Canadian Mathematical Olympiad}\\[1.25in]
%{\LARGE \bf L'Olympiade math\'{e}matique du Canada 2002}
%}}
%\put(1,1){ {\LARGE \bf L'Olympiade math\'{e}matique du Canada}}
%\put(1,0){\LARGE \bf The Canadian Mathematical Olympiad}
%\put(0.4,.6){\bf Wednesday}
%\epsfxsize=1.5in
%\put(2.9,.1){\epsfbox{cmslogo.eps}}
%\put(5.8,.6){{\bf March 27}}
%\end{picture}
%\rule{7 in}{.013in}\\
\noindent
\begin{enumerate}
%\vspace*{0.5cm}
\item
Let $S$ be a subset of $\{1, 2, \dots, 9\}$, such that the sums
formed by adding each unordered pair of distinct numbers from $S$
are all different. For example, the subset $\{1, 2, 3, 5\}$ has
this property, but $\{1, 2, 3, 4, 5\}$ does not, since the pairs
$\{1, 4\}$ and $\{2, 3\}$ have the same sum, namely 5.
What is the maximum number of elements that $S$ can contain?
%\v5
\item
Call a positive integer $n$ {\bf practical} if every positive
integer less than or equal to $n$ can be written as the sum of distinct
divisors of $n$.
\smallskip
For example, the divisors of 6 are {\bf 1}, {\bf 2}, {\bf 3},
and {\bf 6}. Since
\centerline{
1={\bf 1}, ~~ 2={\bf 2}, ~~ 3={\bf 3}, ~~ 4={\bf 1}+{\bf 3}, ~~ 5={\bf 2}+
{\bf 3}, ~~ 6={\bf 6},}
we see that 6 is practical.
Prove that the product of two practical numbers is also practical.
%\v5
\item
Prove that for all positive real numbers $a$, $b$, and $c$,
$$\frac{a^3}{bc} + \frac{b^3}{ca} + \frac{c^3}{ab} \geq a+b+c, $$
and determine when equality occurs.
\item
Let $\Gamma$ be a circle with radius $r$. Let $A$ and $B$ be distinct
points on $\Gamma$ such that $AB < \sqrt{3}r$. Let the circle with centre
$B$ and radius $AB$ meet $\Gamma$ again at $C$. Let $P$ be the point
inside $\Gamma$ such that triangle $ABP$ is equilateral. Finally,
let the line $CP$ meet $\Gamma$ again at $Q$.
Prove that $PQ = r$.
%\v5
\item
Let $N = \{0,1,2,\ldots\}$. Determine all functions
$f: N \rightarrow N$ such that
$$xf(y) + yf(x) = (x+y) f(x^2+y^2)$$
for all $x$ and $y$ in $N$.
\end{enumerate}
\end{document}