25th CanMO 1993


1. Show that there is a unique triangle such that (1) the sides and an altitude have lengths with are 4 consecutive integers, and (2) the foot of the altitude is an integral distance from each vertex.
2. Show that the real number k is rational iff the sequence k, k + 1, k + 2, k + 3, ... contains three (distinct) terms which form a geometric progression.
3. The medians from two vertices of a triangle are perpendicular, show that the sum of the cotangent of the angles at those vertices is at least 2/3.
4. Several schools took part in a tournament. Each player played one match against each player from a different school and did not play anyone from the same school. The total number of boys taking part differed from the total number of girls by 1. The total number of matches with both players of the same sex differed by at most one from the total number of matches with players of opposite sex. What is the largest number of schools that could have sent an odd number of players to the tournament?
5. A sequence of positive integers a₁, a₂, a₃, ... is defined as follows. a₁ = 1, a₂ = 3, a₃ = 2, a_4n = 2a_2n, a_4n+1 = 2a_2n + 1, a_4n+2 = 2a_2n+1 + 1, a_4n+3 = 2a_2n+1. Show that the sequence is a permutation of the positive integers.

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