8th USAMO 1979


1. Find all sets of 14 or less fourth powers which sum to 1599.
2. N is the north pole. A and B are points on a great circle through N equidistant from N. C is a point on the equator. Show that the great circle through C and N bisects the angle ACB in the spherical triangle ABC (a spherical triangle has great circle arcs as sides).
3. a₁, a₂, ... , a_n is an arbitrary sequence of positive integers. A member of the sequence is picked at random. Its value is a. Another member is picked at random, independently of the first. Its value is b. Then a third, value c. Show that the probability that a + b + c is divisible by 3 is at least 1/4.
4. P lies between the rays OA and OB. Find Q on OA and R on OB collinear with P so that 1/PQ + 1/PR is as large as possible.
5. X has n members. Given n+1 subsets of X, each with 3 members, show that we can always find two which have just one member in common.

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