2022 AMC 12A Problem 13

Problem 13 of 25IntermediateGeometry

Let

\mathcal{R}

be the region in the complex plane consisting of all complex numbers

z

that can be written as the sum of complex numbers

z_1

and

z_2,

where

z_1

lies on the segment with endpoints

3

and

4i,

and

z_2

has magnitude at most

1.

What integer is closest to the area of

\mathcal{R}?

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Solution

Adding a disk of radius

1

to every point of the segment sweeps out all points within distance

1

of it. The segment from

3

4i

has length

\sqrt{3^2+4^2}=5.

This “stadium” is a

5\times2

rectangle plus two half-disks of radius

1,

with area

5\cdot2+\pi(1)^2=10+\pi\approx13.14.

The closest integer is

13.

Thus, the correct answer is A.