USAMO 1981

Prove that if $n$ is not a multiple of 3 , then the angle $\pi / n$ can be trisected with ruler and compasses.

What is the largest number of towns that can meet the following criteria. Each pair is directly linked by just one of air, bus or train. At least one pair is linked by air, at least one pair by bus and at least one pair by train. No town has an air link, a bus link and a trian link. No three towns, A, B, C are such that the links between $\mathrm{AB}, \mathrm{AC}$ and $\mathrm{BC}$ are all air, all bus or all train.

Show that for any triangle, $3(\sqrt{3}) / 2 \geq \sin 3 A+\sin 3 B+\sin 3 C \geq-2$. When does equality hold?

A convex polygon has $\mathrm{n}$ sides. Each vertex is joined to a point $\mathrm{P}$ not in the same plane. If $\mathrm{A}, \mathrm{B}, \mathrm{C}$ are adjacent vertices of the polygon take the angle between the planes PBA and PBC. The sum of the $n$ such angles equals the sum of the $\mathrm{n}$ angles subtended at $\mathrm{P}$ by the sides of the polygon (such as the angle APB). Show that $\mathrm{n}=3$.

Show that for any positive real $\mathrm{x},[\mathrm{nx}] \geq \sum_{1}{ }^{\mathrm{n}}[\mathrm{kx}] / \mathrm{k}$.