8th CanMO 1976


1. Given four unequal weights in geometric progression, show how to find the heaviest weight using a balance twice.
2. The real sequence x₀, x₁, x₂, ... is defined by x₀ = 1, x₁ = 2, n(n+1) x_n+1 = n(n-1) x_n - (n-2) x_n-1. Find x₀/x₁ + x₁x₂ + ... + x₅₀/x₅₁.
3. n+2 students played a tournament. Each pair played each other once. A player scored 1 for a win, 1/2 for a draw and nil for a loss. Two students scored a total of 8 and the other players all had equal total scores. Find n.
4. C lies on the segment AB. P is a variable point on the circle with diameter AB. Q lies on the line CP on the opposite side of C to P such that PC/CQ = AC/CB. Find the locus of Q.
5. Show that a positive integer is a sum of two or more consecutive positive integers iff it is not a power of 2.
6. The four points A, B, C, D in space are such that the angles ABC, BCD, CDA, DAB are all right angles. Show that the points are coplanar.
7. p(x, y) is a symmetric polynomial with the factor (x - y). Show that (x - y)² is a factor.
8. A graph has 9 points and 36 edges. Each edge is colored red or blue. Every triangle has a red edge. Show that there are four points with all edges between them red.

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