# Canadian Mathematical Olympiad 1994 

## Problem 1

Evaluate the sum

$$
\sum_{n=1}^{1994}(-1)^{n} \frac{n^{2}+n+1}{n!}
$$

## PROblem 2

Show that every positive integral power of $\sqrt{2}-1$ is of the form $\sqrt{m}-\sqrt{m-1}$ for some positive integer $m$. (e.g. $\left.(\sqrt{2}-1)^{2}=3-2 \sqrt{2}=\sqrt{9}-\sqrt{8}\right)$.

## PROBLEM 3

Twenty-five men sit around a circular table. Every hour there is a vote, and each must respond yes or no. Each man behaves as follows: on the $n^{\text {th }}$ vote, if his response is the same as the response of at least one of the two people he sits between, then he will respond the same way on the $(n+1)^{t h}$ vote as on the $n^{t h}$ vote; but if his response is different from that of both his neighbours on the $n$-th vote, then his response on the $(n+1)$-th vote will be different from his response on the $n^{t h}$ vote. Prove that, however everybody responded on the first vote, there will be a time after which nobody's response will ever change.

## Problem 4

Let $A B$ be a diameter of a circle $\Omega$ and $P$ be any point not on the line through $A$ and $B$. Suppose the line through $P$ and $A$ cuts $\Omega$ again in $U$, and the line through $P$ and $B$ cuts $\Omega$ again in $V$. (Note that in case of tangency $U$ may coincide with $A$ or $V$ may coincide with $B$. Also, if $P$ is on $\Omega$ then $P=U=V$.) Suppose that $|P U|=s|P A|$ and $|P V|=t|P B|$ for some nonnegative real numbers $s$ and $t$. Determine the cosine of the angle $A P B$ in terms of $s$ and $t$.

## Problem 5

Let $A B C$ be an acute angled triangle. Let $A D$ be the altitude on $B C$, and let $H$ be any interior point on $A D$. Lines $B H$ and $C H$, when extended, intersect $A C$ and $A B$ at $E$ and $F$, respectively. Prove that $\angle E D H=\angle F D H$.