19th CanMO 1987


1. Find all positive integer solutions to n! = a² + b² for n < 14.
2. Find all the ways in which the number 1987 can be written in another base as a three digit number with the digits having the same sum 25.
3. ABCD is a parallelogram. X is a point on the side BC such that ACD, ACX and ABX are all isosceles. Find the angles of the parallelogram.
4. n stationary people each fire a water pistol at the nearest person. They are arranged so that the nearest person is always unique. If n is odd, show that at least one person is not hit. Does one person always escape if n is even?
5. Show that [√(4n + 1)] = [√(4n + 2)] = [√(4n + 3)] = [√n + √(n + 1)] for all positive integers n.

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