18th CanMO 1986


1. The triangle ABC has angle B = 90^o. The point D is taken on the ray AC, the other side of C from A, such that CD = AB. ∠CBD = 30^o. Find AC/CD.
2. Three competitors A, B, C compete in a number of sporting events. In each event a points is awarded for a win, b points for second place and c points for third place. There are no ties. The final score was A 22, B 9, C 9. B won the 100 meters. How many events were there and who came second in the high jump?
3. A chord AB of constant length slides around the curved part of a semicircle. M is the midpoint of AB, and C is the foot of the perpendicular from A onto the diameter. Show that angle ACM does not change.
4. Show that (1 + 2 + ... + n) divides (1^k + 2^k + ... + n^k) for k odd.
5. The integer sequence a₁, a₂, a₃, ... is defined by a₁ = 39, a₂ = 45, a_n+2 = a_n+1² - a_n. Show that infinitely many terms of the sequence are divisible by 1986.

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