2022 AMC 12A Problem 13

Question

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}?

13

14

15

16

17

\mathcal{R}

is the set of all points within distance

1

of the segment

The segment has length

5,

so

\mathcal{R}

is a

5\times2

rectangle capped by two half-disks

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

to

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.

Problem 13 in other years