USAMO 1979

Find all sets of 14 or less fourth powers which sum to 1599 .

$\mathrm{N}$ is the north pole. A and $\mathrm{B}$ are points on a great circle through $\mathrm{N}$ equidistant from $\mathrm{N} . \mathrm{C}$ is a point on the equator. Show that the great circle through $\mathrm{C}$ and $\mathrm{N}$ bisects the angle $\mathrm{ACB}$ in the spherical triangle $\mathrm{ABC}$ (a spherical triangle has great circle arcs as sides).

$a_{1}, a_{2}, \ldots, a_{n}$ 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 $\mathrm{a}+\mathrm{b}+\mathrm{c}$ is divisible by 3 is at least $1 / 4$.

$\mathrm{P}$ lies between the rays $\mathrm{OA}$ and $\mathrm{OB}$. Find $\mathrm{Q}$ on $\mathrm{OA}$ and $\mathrm{R}$ on $\mathrm{OB}$ collinear with $\mathrm{P}$ so that $1 / \mathrm{PQ}+1 / \mathrm{PR}$ is as large as possible.

$\mathrm{X}$ has $\mathrm{n}$ members. Given $\mathrm{n}+1$ subsets of $\mathrm{X}$, each with 3 members, show that we can always find two which have just one member in common.