# Difference between revisions of "1982 USAMO Problems"

Problems from the 1982 USAMO.

## Problem 1

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?

## Problem 2

Show that if $m, n$ are positive integers such that $\frac{\left(x^{m+n} + y^{m+n} + z^{m+n}\right)}{m+n}=\left(\frac{x^m + y^m + z^m}{m}\right) \left(\dfrac{x^n + y^n + z^n}{n}}\right)$ (Error compiling LaTeX. ! Extra }, or forgotten \right.) for all real $x, y, z$ with sum $0$, then $(m, n) = (2, 3)$ or $(2, 5)$.

## Problem 3 $D$ is a point inside the equilateral triangle $ABC$. $E$ is a point inside $DBC$. Show that $\frac{\text{area}DBC}{\text{perimeter} DBC^2} > \frac{\text{area} EBC}{\text{perimeter} EBC^2}.$

## Problem 4

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

## Problem 5 $O$ is the center of a sphere $S$. Points $A, B, C$ are inside $S$, $OA$ is perpendicular to $AB$ and $AC$, and there are two spheres through $A, B$, and $C$ which touch $S$. Show that the sum of their radii equals the radius of $S$.