15th USAMO 1986


1. Do there exist 14 consecutive positive integers each divisible by a prime less than 13? What about 21 consecutive positive integers each divisible by a prime less than 17?
2. Five professors attended a lecture. Each fell asleep just twice. For each pair there was a moment when both were asleep. Show that there was a moment when three of them were asleep.
3. What is the smallest n > 1 for which the average of the first n (non-zero) squares is a square?
4. A T-square allows you to construct a straight line through two points and a line perpendicular to a given line through a given point. Circles C and C' intersect at X and Y. XY is a diameter of C. P is a point on C' inside C. Using only a T-square, find points Q,R on C such that QR is perpendicular to XY and PQ is perpendicular to PR.
5. A partition of n is an increasing sequence of integers with sum n. For example, the partitions of 5 are: 1, 1, 1, 1, 1; 1, 1, 1, 2; 1, 1, 3; 1, 4; 5; 1, 2, 2; and 2, 3. If p is a partition, f(p) = the number of 1s in p, and g(p) = the number of distinct integers in the partition. Show that ∑ f(p) = ∑ g(p), where the sum is taken over all partitions of n.

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