USAMO 1984

Two roots of the real quartic $x^{4}-18 x^{3}+a x^{2}+200 x-1984=0$ have product -32 . Find a.

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?

$A, B, C, D, X$ are five points in space, such that $A B, B C, C D, D A$ all subtend the acute angle $\theta$ at $X$. Find the maximum and minimum possible values of $\angle \mathrm{AXC}+\angle \mathrm{BXD}$ (for all such configurations) in terms of $\theta$.

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.

A polynomial of degree $3 n$ has the value 2 at $0,3,6, \ldots, 3 n$, the value 1 at $1,4,7, \ldots, 3 n-2$ and the value 0 at $2,5,8, \ldots, 3 n-1$. Its value at $3 n+1$ is 730 . What is $n$ ?