Problem:
Three mutually tangent spheres of radius rest on a horizontal plane. A sphere of radius rests on them. What is the distance from the plane to the top of the larger sphere?
Answer Choices:
A.
B.
C.
D.
E.
Solution:
Let and be the centers of the three small spheres and the large sphere, respectively. Then is equilateral with side length . If is the intersection of the medians of , then is directly above . Because and , it follows that
Because is unit above the plane and the top of the larger sphere is units above , the distance from the plane to the top of the larger sphere is
Answer: .
The problems on this page are the property of the MAA's American Mathematics Competitions