26th CanMO 1994


1. Find -3/1! + 7/2! - 13/3! + 21/4! - 31/5! + ... + (1994² + 1994 + 1)/1994!
2. Show that every power of (√2 - 1) can be written in the form √(k+1) - √k.
3. 25 people sit in circle. They vote for or against an issue every hour. Each person changes his vote iff his vote was different from both his neighbours on the previous vote. Show that after a while no one's vote changes.
4. AB is the diameter of a circle. C is a point not on the line AB. The line AC cuts the circle again at X and the line BC cuts the circle again at Y. Find cos ACB in terms of CX/CA and CY/CB.
5. ABC is an acute-angled triangle. K is a point inside the triangle on the altitude AD. The line BK meets AC at Y, and the line CK meets AB at Z. Show that ∠ADY = ∠ADZ.

To avoid possible copyright problems, I have changed the wording, but not the substance, of the problems.