15th CanMO 1983


1. Find all solutions to n! = a! + b! + c! .
2. Find all real-valued functions f on the reals whose graphs remain unchanged under all transformations (x, y) → (2^kx, 2^k(kx + y) ), where k is real.
3. Is the volume of a tetrahedron determined by the areas of its faces?
4. Show that we can find infinitely many positive integers n such that 2ⁿ - n is a multiple of any given prime p.
5. Show that the geometric mean of a set S of positive numbers equals the geometric mean of the geometric means of all non-empty subsets of S.

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