2023 AMC 10B Problems/Problem 13: Difference between revisions

Revision as of 17:13, 15 November 2023

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

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

First consider, $|x-1|+|y-1| <= 1.$ We can see that it's a square with radius 1 (diagonal 2). The area of the square is $\sqrt{2}^2 = 2.$

Next, we add one more absolute value and get $|x-1|+||y|-1| <= 1.$ This will double the square reflecting over x-axis.

So now we got 2 squares.

Finally, we add one more absolute value and get $||x|-1|+||y|-1| <= 1.$ This will double the squares reflecting over y-axis.

In the end, we got 4 squares. The total area is $4\cdot2 = 8$ .

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+==Problem 13==
+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>