Problem:
Six congruent circles form a ring with each circle externally tangent to the two circles adjacent to it. All six circles are internally tangent to a circle with radius . Let be the area of the region inside and outside all of the six circles in the ring. Find . (The notation denotes the greatest integer that is less than or equal to .)
Solution:
Let be the radius of each of the six congruent circles, and let and be the centers of two adjacent circles. Join the centers of adjacent circles to form a regular hexagon with side . Let be the center of . Draw the radii of that contain and . Triangle is equilateral, so . Because each of the two radii contains the point where the smaller circle is tangent to , the radius of is , and . The radius of is , so , and .
The problems on this page are the property of the MAA's American Mathematics Competitions