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\begin{document}
{\centerline {\large {\bf 1996 Canadian Mathematical Olympiad}}}
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\begin{enumerate}
\item If $\alpha$, $\beta$, and $\gamma$ are the roots of $x^3 - x - 1 = 0$,
compute $\d \frac{1+\alpha}{1-\alpha} + \frac{1+\beta}{1-\beta} +
\frac{1+\gamma}{1-\gamma}$.

\item Find all real solutions to the following system of equations.  Carefully
justify your answer.
$$\left\{
\begin{array}{c}
\d \frac{4x^2}{1+4x^2} = y \\
\\
\d \frac{4y^2}{1+4y^2} = z \\
\\
\d \frac{4z^2}{1+4z^2} = x
\end{array}
\right. $$

\item We denote an arbitrary permutation of the integers 1, 2, \ldots, $n$ by
$a_1$, $a_2$, \ldots, $a_n$.  Let $f(n)$ denote the number of these
permutations such that:
\begin{enumerate}
\item[(i)] $a_1 = 1$;
\item[(ii)] $|a_i - a_{i+1}| \leq 2$, \ $i = 1, \ldots, n - 1$.
\end{enumerate}
Determine whether $f(1996)$ is divisible by 3.

\item Let triangle $ABC$ be an isosceles triangle with $AB = AC$.  Suppose that
the angle bisector of $\angle B$ meets $AC$ at $D$ and that $BC = BD + AD$.
Determine $\angle A$.

\item Let $r_1$, $r_2$, \ldots, $r_m$ be a given set of $m$ positive rational
numbers such that $\d \sum_{k=1}^m r_k = 1$.  Define the function $f$ by $f(n)
= n - \d \sum_{k=1}^m \: [r_k n]$ for each positive integer $n$.  Determine the
minimum and maximum values of $f(n)$.  Here $[x]$ denotes the greatest integer
less than or equal to $x$.
\end{enumerate}
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