16th CanMO 1984


1. Show that the sum of 1984 consecutive positive integers cannot be a square.
2. You have keyring with n identical keys. You wish to color code the keys so that you can distinguish them. What is the smallest number of colors you need? [For example, you could use three colors for eight keys: R R R R G B R R. Starting with the blue key and moving away from the green key uniquely distinguishes each of the red keys.]
3. Show that there are infinitely many integers which have no zeros and which are divisible by the sum of their digits.
4. An acute-angled triangle has unit area. Show that there is a point inside the triangle which is at least 2/(3^3/4) from any vertex.
5. Given any seven real numbers show we can select two, x and y, such that 0 ≤ (x - y)/(1 + xy) ≤ 1/√3.

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