USAMO 1982

A graph has 1982 points. Given any four points, there is at least one joined to the other three. What is the smallest number of points which are joined to 1981 points?

Show that if $m, n$ are positive integers such that $\left(x^{m+n}+y^{m+n}+z^{m+n}\right) /(m+n)=\left(x^{m}+y^{m}+z^{m}\right) / m\left(x^{n}+y^{n}+\right.$ $\left.\mathrm{z}^{\mathrm{n}}\right) / \mathrm{n}$ for all real $\mathrm{x}, \mathrm{y}, \mathrm{z}$ with sum 0 , then $\{\mathrm{m}, \mathrm{n}\}=\{2,3\}$ or $\{2,5\}$.

$\mathrm{D}$ is a point inside the equilateral triangle $\mathrm{ABC}$. $\mathrm{E}$ is a point inside $\mathrm{DBC}$. Show that area $\mathrm{DBC} /($ perimeter $\mathrm{DBC})^{2}>$ area $\mathrm{EBC} /(\text { perimeter } \mathrm{EBC})^{2}$.

Show that there is a positive integer $\mathrm{k}$ such that, for every positive integer $\mathrm{n}, \mathrm{k} 2^{\mathrm{n}}+1$ is composite.

$\mathrm{O}$ is the center of a sphere $\mathrm{S}$. Points $\mathrm{A}, \mathrm{B}, \mathrm{C}$ are inside $\mathrm{S}, \mathrm{OA}$ is perpendicular to $\mathrm{AB}$ and $\mathrm{AC}$, and there are two spheres through $A, B$, and $\mathrm{C}$ which touch $\mathrm{S}$. Show that the sum of their radii equals the radius of $\mathrm{S}$.