34th CanMO 2002


1. What is the largest possible number of elements in a subset of {1, 2, 3, ... , 9} such that the sum of every pair (of distinct elements) in the subset is different?
2. We say that the positive integer m satisfies condition X if every positive integer less than m is a sum of distinct divisors of m. Show that if m and n satisfy condition X, then so does mn.
3. Show that x³/(yz) + y³/(zx) + z³/(xy) ≥ x + y + z for any positive reals x, y, z. When do we have equality?
4. ABC is an equilateral triangle. C lies inside a circle center O through A and B. X and Y are points on the circle such that AB = BX and C lies on the chord XY. Show that CY = AO.
5. Let X be the set of non-negative integers. Find all functions f: X → X such that x f(y) + y f(x) = (x + y) f(x² + y²) for all x, y.

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