Problem:
Three mutually tangent spheres of radius 1 rest on a horizontal plane. A sphere of radius 2 rests on them. What is the distance from the plane to the top of the larger sphere?
Answer Choices:
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Solution:
Let and be the centers of the three small spheres and the large sphere, respectively. Then is equilateral with side length 2 . If is the intersection of the medians of , then is directly above . Because and , it follows that
Because is 1 unit above the plane and the top of the larger sphere is 2 units above , the distance from the plane to the top of the larger sphere is
The problems on this page are the property of the MAA's American Mathematics Competitions