2023 AMC 12B Problem 9

Attempt Problem 9 of the 2023 AMC 12B below, then check your answer against the professionally curated solution 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.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

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

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

$2$

$8$

$4$

$15$

$12$

Answer: B

Concepts:absolute value area symmetry

Difficulty rating: 1500

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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