22nd CanMO 1990


1. A competition is played amongst n > 1 players over d days. Each day one player gets a score of 1, another a score of 2, and so on up to n. At the end of the competition each player has a total score of 26. Find all possible values for (n, d).
2. n(n + 1)/2 distinct numbers are arranged at random into n rows. The first row has 1 number, the second has 2 numbers, the third has 3 numbers and so on. Find the probability that the largest number in each row is smaller than the largest number in each row with more numbers.
3. The feet of the perpendiculars from the intersection point of the diagonals of a convex cyclic quadrilateral to the sides form a quadrilateral q. Show that the sum of the lengths of each pair of opposite sides of q is equal.
4. A particle can travel at a speed of 2 meters/sec along the x-axis and 1 meter/sec elsewhere. Starting at the origin, which regions of the plane can the particle reach within 1 second.
5. N is the positive integers, R is the reals. The function f : N → R satisfies f(1) = 1, f(2) = 2 and f(n+2) = f(n+2 - f(n+1) ) + f(n+1 - f(n) ). Show that 0 ≤ f(n+1) - f(n) ≤ 1. Find all n for which f(n) = 1025.

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