4th USAMO 1975


1. Show that for any non-negative reals x, y, [5x] + [5y] ≥ [3x+y] + [x+3y]. Hence or otherwise show that (5a)! (5b)!/( a! b! (3a+b)! (a+3b)! ) is integral for any positive integers a, b.
2. Show that for any tetrahedron the sum of the squares of the lengths of two opposite edges is at most the sum of the squares of the other four.
3. A polynomial p(x) of degree n satisfies p(0) = 0, p(1) = 1/2, p(2) = 2/3, ... , p(n) = n/(n+1). Find p(n+1).
4. Two circles intersect at two points, one of them X. Find Y on one circle and Z on the other, so that X, Y and Z are collinear and XY.XZ is as large as possible.
5. A pack of n cards, including three aces, is well shuffled. Cards are turned over in turn. Show that the expected number of cards that must be turned over to reach the second ace is (n+1)/2.

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