2005 AIME I Problems/Problem 1
Six congruent circles form a ring with each circle externally tangent to two circles adjacent to it. All circles are internally tangent to a circle with radius 30. Let be the area of the region inside circle and outside of the six circles in the ring. Find (the floor function).
Define the radii of the six congruent circles as . If we draw all of the radii to the points of external tangency, we get a regular hexagon. If we connect the vertices of the hexagon to the center of the circle , we form several equilateral triangles. The length of each side of the triangle is . Notice that the radius of circle is equal to the length of the side of the triangle plus . Thus, the radius of has a length of , and so . , so .
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