1992 USAMO Problems/Problem 2
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Contents
Problem
Prove
Solution 1
Consider the points in the coordinate plane with origin
, for integers
.
Evidently, the angle between segments and
is
, and the length of segment
is
. It then follows that the area of triangle
is
. Therefore
so
as desired.
Solution 2
First multiply both sides of the equation by , so the right hand side is
. Now by rewriting
, we can derive the identity
. Then the left hand side of the equation simplifies to
as desired.
Alternate solutions are always welcome. If you have a different, elegant solution to this problem, please add it to this page.
Resources
1992 USAMO (Problems • Resources) | ||
Preceded by Problem 1 |
Followed by Problem 3 | |
1 • 2 • 3 • 4 • 5 | ||
All USAMO Problems and Solutions |