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{\Large \bf THE 2002 CANADIAN MATHEMATICAL OLYMPIAD}
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\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}

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