2021 AMC 10B Spring Problem 7

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

Circles on the same side of the line are nested

Inside exactly one nested circle means outside the next smaller circle

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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

\begin{aligned} &7^2\pi+(5^2-3^2)\pi \\ &=49\pi+16\pi=65\pi. \end{aligned}

Thus, the answer is D .

Problem 7 in other years