1982 USAMO Problems

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?

Solution

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. Unknown error_msg) for all real $x, y, z$ with sum $0$ , then $(m, n) = (2, 3)$ or $(2, 5)$ .

Solution

Problem 3

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

Solution

Problem 4

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

Solution

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 $S$ .

Solution

1982 USAMO (Problems • Resources)
Preceded by 1981 USAMO	Followed by 1983 USAMO
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1982 USAMO Problems

Contents

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

See Also