35th CanMO 2003


1. The angle between the hour and minute hands of a standard 12-hour clock is exactly 1^o. The time is an integral number n of minutes after noon (where 0 < n < 720). Find the possible values of n.
2. What are the last three digits of 2003^N, where N = 2002²⁰⁰¹.
3. Find all positive real solutions to x³ + y³ + z³ = x + y + z, x² + y² + z² = xyz.
4. Three fixed circles pass through the points A and B. X is a variable point on the first circle different from A and B. The line AX meets the other two circles at Y and Z (with Y between X and Z). Show that XY/YZ is independent of the position of X.
5. S is any set of n distinct points in the plane. The shortest distance between two points of S is d. Show that there is a subset of at least n/7 points such that each pair is at least a distance d√3 apart.

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