Problem:
Eight spheres of radius are placed on a flat surface so that each sphere is tangent to two others and their centers are the vertices of a regular octagon. A ninth sphere is placed on the flat surface so that it is tangent to each of the other eight spheres. The radius of this last sphere is , where , and are positive integers, and is not divisible by the square of any prime. Find .
Solution:
Let be the distance between the center of one of the eight congruent spheres and the center of the regular octagon. Apply the Law of Cosines to the isosceles triangle formed by the center of the octagon and the centers of two congruent tangent spheres (shown below in the top view). This yields , from which follows
Let be the radius of the ninth sphere. The center of one of the eight congruent spheres, the center of the octagon, and the center of the ninth sphere form a right triangle (shown below in the side view). Apply the Pythagorean Theorem to obtain , which is equivalent to . It follows that , and thus .
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The problems on this page are the property of the MAA's American Mathematics Competitions