2022 AMC 10 B

Complete problem set with solutions and individual problem pages

Problem 22 Hard

Let S be the set of circles in the coordinate plane that are tangent to each of the three circles with equations x^2+y^2=4, x^2+y^2=64 and \left( x-5\right)^{2}+y^{2}=3. What is sum of the areas of all circles in S.

  • A.

    48\pi

  • B.

    68\pi

  • C.

    96\pi

  • D.

    102\pi

  • E.

    136\pi

Answer:E

There are two circles that are externally tangent to the two small circles and internally tangent to the large circle. Here r=3 and A=18\pi. (blue part)

There are two circles that are internally tangent to the concentric circles and externally tangent to the other circle. Here r=5 and A=50\pi. (red part)

There are two circles that are internally tangent to the big and non-center circles, but externally tangent to the small center circle. Here r=3 and A=18\pi. (green part)

There are two circles that are internally tangent to all three circles. Here r=5 and A=50\pi. (brown part)

The sum of area of all circles is 136\pi.

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