# 2005 AIME I Problems/Problem 1

## Problem

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 $C$ with radius 30. Let $K$ be the area of the region inside circle $C$ and outside of the six circles in the ring. Find $\lfloor K \rfloor$ (the floor function).

## Solution

Define the radii of the six congruent circles as $r$. 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 $C$, we form several equilateral triangles. The length of each side of the triangle is $2r$. Notice that the radius of circle $C$ is equal to the length of the side of the triangle plus $r$. Thus, the radius of $C$ has a length of $3r = 30$, and so $r = 10$. $K = 30^2\pi - 6(10^2\pi) = 300\pi$, so $\lfloor 300\pi \rfloor = \boxed{942}$.

## See also

 2005 AIME I (Problems • Answer Key • Resources) Preceded byFirst Question Followed byProblem 2 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 All AIME Problems and Solutions

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