CMO 1994

Evaluate $\sum_{n=1}^{1994}{\left((-1)^{n}\cdot\left(\frac{n^2 + n + 1}{n!}\right)\right)}$ .

Prove that $(\sqrt{2}-1)^n$ $\forall n\in \mathbb{Z}^{+}$ can be represented as $\sqrt{m}-\sqrt{m-1}$ for some $m\in \mathbb{Z}^{+}$ .

$25$ men sit around a circular table. Every hour there is a vote, and each must respond yesor 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 $(n+1)^{\text{th}}$ vote as on the $n^{\text{th}}$ vote; but if his response is different from that of both his neighbours on the $n^{\text{th}}$ vote, then his response on the $(n+1)^{\text{th}}$ vote will be different from his response on the $n^{\text{th}}$ vote. Prove that, however everybody responded on the first vote, there will be a time after which nobody's response will ever change.

Let $AB$ be a diameter of a circle $\Omega$ and $P$ be any point not on the line through $AB$ . Suppose that the line through $PA$ cuts $\Omega$ again at $U$ , and the line through $PB$ cuts $\Omega$ at $V$ . Note that in case of tangency, $U$ may coincide with $A$ or $V$ might coincide with $B$ . Also, if $P$ is on $\Omega$ then $P=U=V$ . Suppose that $|PU|=s|PA|$ and $|PV|=t|PB|$ for some $0\le s,t\in \mathbb{R}$ . Determine $\cos \angle APB$ in terms of $s,t$ .

Let $ABC$ be an acute triangle. Let $AD$ be the altitude on $BC$ , and let $H$ be any interior point on $AD$ . Lines $BH,CH$ , when extended, intersect $AC,AB$ at $E,F$ respectively. Prove that $\angle EDH=\angle FDH$ .