# Canadian Mathematical Olympiad 

## PROBLEM 1

In the diagram, $O B_{i}$ is parallel and equal in length to $A_{i} A_{i+1}$ for $i=1,2,3$ and $4\left(A_{5}=A_{1}\right)$. Show that the area of $B_{1} B_{2} B_{3} B_{4}$ is twice that of $A_{1} A_{2} A_{3} A_{4}$.
![](https://cdn.mathpix.com/cropped/2024_11_22_36f5c7fa524e4a9b6649g-1.jpg?height=538&width=484&top_left_y=723&top_left_x=731)

PROblem 2
If $a, b$ and $c$ are the roots of the equation $x^{3}-x^{2}-x-1=0$,
(i) show that $a, b$ and $c$ are distinct:
(ii) show that

$$
\frac{a^{1982}-b^{1982}}{a-b}+\frac{b^{1982}-c^{1982}}{b-c}+\frac{c^{1982}-a^{1982}}{c-a}
$$

is an integer.

## Problem 3

Let $R^{n}$ be the $n$-dimensional Euclidean space. Determine the smallest number $g(n)$ of points of a set in $R^{n}$ such that every point in $R^{n}$ is at irrational distance from at least one point in that set.

## PROBLEM 4

Let $p$ be a permutation of the set $S_{n}=\{1,2, \ldots, n\}$. An element $j \in S_{n}$ is called a fixed point of $p$ if $p(j)=j$. Let $f_{n}$ be the number of permutations having no fixed points, and $g_{n}$ be the number with exactly one fixed point. Show that $\left|f_{n}-g_{n}\right|=1$.

Problem 5
The altitudes of a tetrahedron $A B C D$ are extended externally to points $A^{\prime}, B^{\prime}$, $C^{\prime}$ and $D^{\prime}$ respectively, where $A A^{\prime}=k / h_{a}, B B^{\prime}=k / h_{b}, C C^{\prime}=k / h_{c}$ and $D D^{\prime}=$
$k / h_{d}$. Here, $k$ is a constant and $h_{a}$ denotes the length of the altitude of $A B C D$ from vertex $A$, etc. Prove that the centroid of the tetrahedron $A^{\prime} B^{\prime} C^{\prime} D^{\prime}$ coincides with the centroid of $A B C D$.