1982 USAMO Problems/Problem 5
Problem
, and are three interior points of a sphere such that and are perpendicular to the diameter of through , and so that two spheres can be constructed through , , and which are both tangent to . Prove that the sum of their radii is equal to the radius of .
Solution
Let the two tangent spheres be and , and let and be the origins and radii of respectively. Then stands normal to the plane through . Because both spheres go through , , and , the line also stands normal to , meaning and are both coplanar and parallel. Therefore the problem can be flattened to the plane through , , and .
Let be points on such that
Let be the three circles radical center, meaning and are tangent segments to and .
Because we have that and are diameters.
This means that and .
And because and we have that and .
We then conclude that
Let denote line bisecting line segment .
Since it follows that . Similarly we have that .
And so is a parallelogram because and bisect each other, meaning
See Also
1982 USAMO (Problems • Resources) | ||
Preceded by Problem 4 |
Followed by Last Question | |
1 • 2 • 3 • 4 • 5 | ||
All USAMO Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.