13th CanMO 1981


1. Show that there are no real solutions to [x] + [2x] + [4x] + [8x] + [16x] + [32x] = 12345.
2. The circle C has radius 1 and touches the line L at P. The point X lies on C and Y is the foot of the perpendicular from X to L. Find the maximum possible value of area PXY (as X varies).
3. Given a finite set of lines in the plane, show that an arbitrarily large circle can be drawn which does not meet any of them. Show that there is a countable set of lines in the plane such that any circle with positive radius meets at least one of them.
4. p(x) and q(x) are real polynomials such that p(q(x) ) = q(p(x) ) and p(x) = q(x) has no real solutions. Show that p(p(x) ) = q(q(x) ) has no real solutions.
5. 11 groups perform at a festival. Each day any groups not performing watch the others (but groups performing that day do not watch the others). What is the smallest number of days for which the festival can last if every group watches every other group at least once during the festival?

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