2023 AMC 12B Problem 9

Below is the professionally curated solution for Problem 9 of the 2023 AMC 12B, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2023 AMC 12B solutions, or check the answer key.

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Concepts:absolute value area symmetry

Difficulty rating: 1500

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

$||x|-1|+||y|-1|\le 1?$

$2$

$8$

$4$

$15$

$12$

Solution:

Replacing $x,y$ by $|x|,|y|,$ the condition $|x-1|+|y-1|\le 1$ describes a diamond centered at $(1,1)$ with diagonals of length $2,$ hence area $2.$ It lies entirely in the first quadrant (touching the axes only at single points), so reflecting across the two axes produces $4$ disjoint copies. The total area is $4\times 2=8.$

Thus, the correct answer is B.

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