Question

In a plane, four circles with radii $1,3,5,$ and $7$ are tangent to line $\ell$ at the same point $A,$ but they may be on either side of $\ell.$ Region $S$ consists of all the points that lie inside exactly one of the four circles. What is the maximum possible area of region $S?$

$24\pi$

$32\pi$

$64\pi$

$65\pi$

$84\pi$

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Written solution:

On one side of $\ell$ , circles tangent at $A$ are nested. For nested circles with radii $r_1>r_2>\cdots$ , the points inside exactly one of those circles have area $\pi(r_1^2-r_2^2)$ if there are at least two circles; a third smaller nested circle does not count because its points are inside three circles, not exactly one.

To maximize the area, put the circle of radius $7$ alone on one side, and put the circles of radii $5,3,1$ on the other side. This gives

$7^2\pi+(5^2-3^2)\pi=49\pi+16\pi=65\pi.$

Thus, the answer is D .

2021 AMC 10B Spring Problem 7

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Problem 7 in Other Years