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| − | ==Problem 13==
| + | #redirect[[2023 AMC 12B Problems/Problem 9]] |
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| − | What is the area of the region in the coordinate plane defined by
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| − | <math>| | x | - 1 | + | | y | - 1 | \le 1</math>?
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| − | <math>\text{(A)}\ 2 \qquad \text{(B)}\ 8 \qquad \text{(C)}\ 4 \qquad \text{(D)}\ 15 \qquad \text{(E)}\ 12</math>
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| − | == Solution ==
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| − | First consider, <math>|x-1|+|y-1| <= 1.</math>
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| − | We can see that it's a square with radius 1 (diagonal 2). The area of the square is <math>\sqrt{2}^2 = 2.</math>
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| − | Next, we add one more absolute value and get <math>|x-1|+||y|-1| <= 1.</math> This will double the square reflecting over x-axis.
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| − | So now we got 2 squares.
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| − | Finally, we add one more absolute value and get <math>||x|-1|+||y|-1| <= 1.</math> This will double the squares reflecting over y-axis.
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| − | In the end, we got 4 squares. The total area is <math>4\cdot2 = 8</math>.
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