2023 AMC 10B Problem 13

Question

What is the area of the region in the coordinate plane defined by the inequality

\bigl||x| - 1\bigr| + \bigl||y| - 1\bigr| \le 1?

2

8

4

15

12

Substitute

u = |x|

and

v = |y|;

then

|u - 1| + |v - 1| \le 1

is a diamond

Reflecting the first-quadrant piece across both axes copies it

4

times

Substitute

u = |x|,

v = |y|.

Then

|u - 1| + |v - 1| \le 1

is a diamond centered at

(1, 1)

with diagonals of length

2,

so it has area

2,

and it sits entirely in

u, v \ge 0.

The map

(x, y) \mapsto (|x|, |y|)

is four-to-one over

u, v \gt 0,

so the full region has area

4 \times 2 = 8.

Thus, B is the correct answer.

Problem 13 in other years