Difference between revisions of "1982 USAMO Problems"

Revision as of 13:35, 17 September 2012

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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 ==Problem 5==
 <math>O</math> is the center of a sphere <math>S</math>. Points <math>A, B, C</math> are inside <math>S</math>, <math>OA</math> is perpendicular to <math>AB</math> and <math>AC</math>, and there are two spheres through <math>A, B</math>, and <math>C</math> which touch <math>S</math>.  Show that the sum of their radii equals the radius of <math>S</math>.
+== See Also ==
+{{USAMO box|year=1982|before=[[1981 USAMO]]|after=[[1983 USAMO]]}}

1982 USAMO (Problems • Resources)
Preceded by 1981 USAMO	Followed by 1983 USAMO
1 • 2 • 3 • 4 • 5
All USAMO Problems and Solutions

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