Difference between revisions of "2023 AMC 10B Problems/Problem 13"

Latest revision as of 19:36, 15 November 2023

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-==Problem 13==
+#redirect[[2023 AMC 12B Problems/Problem 9]]
-What is the area of the region in the coordinate plane defined by
-<math>| | x | - 1 | + | | y | - 1 | \le 1</math>?
-== Solution ==
-First consider, <math>|x-1|+|y-1| <= 1.</math>
-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>
-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.
-So now we got 2 squares.
-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.
-In the end, we got 4 squares.  The total area is <math>4\cdot2 = 8</math>.