11th CanMO 1979


1. If a > b > c > d is an arithmetic progression of positive reals and a > h > k > d is a geometric progression of positive reals, show that bc ≥ hk.
2. Show that two tetrahedra do not necessarily have the same sum for their dihedral angles.
3. Given five distinct integers greater than one, show that the sum of the inverses of the four lowest common multiples of the adjacent pairs is at most 15/16. [Two of the numbers are adjacent if none of the others lies between them.]
4. A dog is standing at the center of a circular yard. A rabbit is at the edge. The rabbit runs round the edge at constant speed v. The dog runs towards the rabbit at the same speed v, so that it always remains on the line between the center and the rabbit. Show that it reaches the rabbit when the rabbit has run one quarter of the way round.
5. The lattice is the set of points (x, y) in the plane with at least one coordinate integral. Let f(n) be the number of walks of n steps along the lattice starting at the origin. Each step is of unit length from one intersection point to another. We only count walks which do not visit any point more than once. Find f(n) for n =1, 2, 3, 4 and show that 2ⁿ < f(n) ≤ 4·3^n-1.

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