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2023 AMC 12B Problem 9

Problem 9 of 25EasierAlgebraGeometry

What is the area of the region in the coordinate plane defined by ∣∣x∣−1∣+∣∣y∣−1∣≤1? ||x|-1|+||y|-1|\le 1?

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Solution

Replacing x,yx,y by ∣x∣,∣y∣,|x|,|y|, the condition ∣x−1∣+∣y−1∣≤1|x-1|+|y-1|\le 1 describes a diamond centered at (1,1)(1,1) with diagonals of length 2,2, hence area 2.2. It lies entirely in the first quadrant (touching the axes only at single points), so reflecting across the two axes produces 44 disjoint copies. The total area is 4×2=8.4\times 2=8. Thus, the correct answer is B.

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Concepts: absolute value · area · symmetry

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.