# 1982 USAMO Problems

Problems from the 1982 USAMO.

## Problem 1

In a party with $1982$ persons, among any group of four there is at least one person who knows each of the other three. What is the minimum number of people in the party who know everyone else.

## 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 $S$.

## See Also

 1982 USAMO (Problems • Resources) Preceded by1981 USAMO Followed by1983 USAMO 1 • 2 • 3 • 4 • 5 All USAMO Problems and Solutions
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