1982 USAMO Problems

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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 $\frac{\left(x^{m+n} + y^{m+n} + z^{m+n}\right)}{(m+n)} =\frac{ (x^m + y^m + z^m)}{\frac{m \left(x^n + y^n + z^n\right)}{n}}$ 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 $S$.

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