# Difference between revisions of "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

Let $S_r=x^r+y^r+z^r$ with $x,y,z$ real. It is known that if $S_1=0$,

$(*)$ $\frac{S_{m+n}}{m+n}=\frac{S_m}{m}\frac{S_n}{n}$

for $(m,n)=(2,3),(3,2),(2,5)$, or $(5,2)$. Determine all other pairs of integers $(m,n)$ if any, so that $(*)$ holds for all real numbers $x,y,z$ such that $S_1=0$.

## Problem 3

If a point $A_1$ is in the interior of an equilateral triangle $ABC$ and point $A_2$ is in the interior of $\triangle{A_1BC}$, prove that

$I.Q. (A_1BC) > I.Q.(A_2BC)$,

where the isoperimetric quotient of a figure $F$ is defined by

$I.Q.(F) = \frac{\text{Area (F)}}{\text{[Perimeter (F)]}^2}$

## Problem 4

Prove that there exists a positive integer $k$ such that $k\cdot2^n+1$ is composite for every positive integer $n$.

## Problem 5

$A,B$, and $C$ are three interior points of a sphere $S$ such that $AB$ and $AC$ are perpendicular to the diameter of $S$ through $A$, and so that two spheres can be constructed through $A$, $B$, and $C$ which are both tangent to $S$. Prove that the sum of their radii is equal to the radius of $S$.

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