31st CanMO 1999


1. Find all real solutions to the equation 4x² - 40[x] + 51 = 0.
2. ABC is equilateral. A circle with center on the line through A parallel to BC touches the segment BC. Show that the length of arc of the circle inside ABC is independent of the position of the circle.
3. Find all positive integers which equal the square of their number of positive divisors.
4. X is a subset of eight elements of {1, 2, 3, ... , 17}. Show that there are three pairs of (distinct) elements with the same difference.
5. x, y, z are non-negative reals with sum 1, show that x²y + y²z + z²x ≤ 4/27. When do we have equality?

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