13th USAMO 1984


1. Two roots of the real quartic x⁴ - 18x³ + ax² + 200x - 1984 = 0 have product -32. Find a.
2. Can one find a set of n distinct positive integers such that the geometric mean of any (non-empty, finite) subset is an integer? Can one find an infinite set with this property?
3. A, B, C, D, X are five points in space, such that AB, BC, CD, DA all subtend the acute angle θ at X. Find the maximum and minimum possible values of ∠AXC + ∠BXD (for all such configurations) in terms of θ.
4. A maths exam has two papers, each with at least one question and 28 questions in total. Each pupil attempted 7 questions. Each pair of questions was attempted by just two pupils. Show that one pupil attempted either nil or at least 4 questions in the first paper.
5. A polynomial of degree 3n has the value 2 at 0, 3, 6, ... , 3n, the value 1 at 1, 4, 7, ... , 3n-2 and the value 0 at 2, 5, 8, ... , 3n-1. Its value at 3n+1 is 730. What is n?

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