Problem:
Six spheres of radius 1 are positioned so that their centers are at the vertices of a regular hexagon of side length 2 . The six spheres are internally tangent to a larger sphere whose center is the center of the hexagon. An eighth sphere is externally tangent to the six smaller spheres and internally tangent to the larger sphere. What is the radius of this eighth sphere?
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Solution:
Let the vertices of the regular hexagon be labeled in order , , and . Let be the center of the hexagon, which is also the center of the largest sphere. Let the eighth sphere have center and radius . Because the centers of the six small spheres are each a distance 2 from and the small spheres have radius 1 , the radius of the largest sphere is 3 . Because
is equidistant from and , the segments and are perpendicular. Let be the distance from to . Then . The Pythagorean Theorem applied to gives , which simplifies to , so . Note that this shows that the eighth sphere is tangent to at .
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The problems on this page are the property of the MAA's American Mathematics Competitions