# Canadian Mathematical Olympiad 1986 

## Problem 1

In the diagram line segments $A B$ and $C D$ are of length 1 while angles $A B C$ and $C B D$ are $90^{\circ}$ and $30^{\circ}$ respectively. Find $A C$.
![](https://cdn.mathpix.com/cropped/2024_11_22_0910fe4f2058cd7b2b17g-1.jpg?height=304&width=589&top_left_y=710&top_left_x=692)

## PROBLEM 2

A Mathlon is a competition in which there are $M$ athletic events. Such a competition was held in which only $A, B$, and $C$ participated. In each event $p_{1}$ points were awarded for first place, $p_{2}$ for second and $p_{3}$ for third, where $p_{1}>p_{2}>p_{3}>0$ and $p_{1}, p_{2}, p_{3}$ are integers. The final score for $A$ was 22 , for $B$ was 9 and for $C$ was also 9. $B$ won the 100 metres. What is the value of $M$ and who was second in the high jump?

## Problem 3

A chord $S T$ of constant length slides around a semicircle with diameter $A B . M$ is the mid-point of $S T$ and $P$ is the foot of the perpendicular from $S$ to $A B$. Prove that angle $S P M$ is constant for all positions of $S T$.

## Problem 4

For positive integers $n$ and $k$, define $F(n, k)=\sum_{r=1}^{n} r^{2 k-1}$. Prove that $F(n, 1)$ divides $F(n, k)$.

Problem 5
Let $u_{1}, u_{2}, u_{3}, \ldots$ be a sequence of integers satisfying the recurrence relation $u_{n+2}=$ $u_{n+1}^{2}-u_{n}$. Suppose $u_{1}=39$ and $u_{2}=45$. Prove that 1986 divides infinitely many terms of the sequence.