20th CanMO 1988


1. For what real values of k do 1988x² + kx + 8891 and 8891x² + kx + 1988 have a common zero?
2. Given a triangle area A and perimeter p, let S be the set of all points a distance 5 or less from a point of the triangle. Find the area of S.
3. Given n > 4 points in the plane, some of which are colored red and the rest black. No three points of the same color are collinear. Show that we can find three points of the same color, such that two of the points do not have a point of the opposite color on the segment joining them.
4. Define two integer sequences a₀, a₁, a₂, ... and b₀, b₁, b₂, ... as follows. a₀ = 0, a₁ = 1, a_n+2 = 4a_n+1 - a_n, b₀ = 1, b₁ = 2, b_n+2 = 4b_n+1 - b_n. Show that b_n² = 3a_n² + 1.
5. If S is a sequence of positive integers let p(S) be the product of the members of S. Let m(S) be the arithmetic mean of p(T) for all non-empty subsets T of S. S' is formed from S by appending an additional positive integer. If m(S) = 13 and m(S') = 49, find S'.

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