# Canadian Mathematical Olympiad 1996 

## Problem 1

If $\alpha, \beta, \gamma$ are the roots of $x^{3}-x-1=0$, compute

$$
\frac{1+\alpha}{1-\alpha}+\frac{1+\beta}{1-\beta}+\frac{1+\gamma}{1-\gamma}
$$

## PROBLEM 2

Find all real solutions to the following system of equations. Carefully justify your answer.

$$
\left\{\begin{array}{l}
\frac{4 x^{2}}{1+4 x^{2}}=y \\
\frac{4 y^{2}}{1+4 y^{2}}=z \\
\frac{4 z^{2}}{1+4 z^{2}}=x
\end{array}\right.
$$

PROBLEM 3
We denote an arbitrary permutation of the integers $1, \ldots, n$ by $a_{1}, \ldots, a_{n}$. Let $f(n)$ be the number of these permutations such that
(i) $a_{1}=1$;
(ii) $\left|a_{i}-a_{i+1}\right| \leq 2, \quad i=1, \ldots, n-1$.

Determine whether $f(1996)$ is divisible by 3 .

## PROblem 4

Let $\triangle A B C$ be an isosceles triangle with $A B=A C$. Suppose that the angle bisector of $\angle B$ meets $A C$ at $D$ and that $B C=B D+A D$. Determine $\angle A$.

Problem 5
Let $r_{1}, r_{2}, \ldots, r_{m}$ be a given set of $m$ positive rational numbers such that $\sum_{k=1}^{m} r_{k}=1$. Define the function $f$ by $f(n)=n-\sum_{k=1}^{m}\left[r_{k} n\right]$ 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 $x$