USAMO 1983

If six points are chosen sequentially at random on the circumference of a circle, what is the probability that the triangle formed by the first three is disjoint from that formed by the second three.

Show that the five roots of the quintic $a_{5} x^{5}+a_{4} x^{4}+a_{3} x^{3}+a_{2} x^{2}+a_{1} x+a_{0}=0$ are not all real if $2 a_{4}{ }^{2}<5 a_{5} a_{3}$.

$\mathrm{S}_{1}, \mathrm{~S}_{2}, \ldots, \mathrm{S}_{\mathrm{n}}$ are subsets of the real line. Each $\mathrm{S}_{\mathrm{i}}$ is the union of two closed intervals. Any three $\mathrm{S}_{\mathrm{i}}$ have a point in common. Show that there is a point which belongs to at least half the $\mathrm{S}_{\mathrm{i}}$.

Show that one can construct (with ruler and compasses) a length equal to the altitude from $\mathrm{A}$ of the tetrahedron $A B C D$, given the lengths of all the sides. [So for each pair of vertices, one is given a pair of points in the plane the appropriate distance apart.]

Prove that an open interval of length $1 / \mathrm{n}$ in the real line contains at most $(\mathrm{n}+1) / 2$ rational points $\mathrm{p} / \mathrm{q}$ with $1 \leq$ $\mathrm{q} \leq \mathrm{n}$.