2011 AMC 12A Problem 18

Below is the professionally curated solution for Problem 18 of the 2011 AMC 12A, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2011 AMC 12A solutions, or check the answer key.

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Concepts:absolute value optimization square (geometry)

Difficulty rating: 1840

18.

Suppose that $|x + y| + |x - y| = 2.$ What is the maximum possible value of $x^2 - 6x + y^2?$

$5$

$6$

$7$

$8$

$9$

Solution:

The identity $|x+y| + |x-y| = 2\max(|x|, |y|)$ turns the condition into $\max(|x|, |y|) = 1,$ the boundary of the square with $|x| \le 1$ and $|y| \le 1.$

On this region $x^2 - 6x + y^2$ increases as $x$ decreases and as $y^2$ increases, so the maximum is at $x = -1,$ $y = \pm 1:$ $1 + 6 + 1 = 8.$

Thus, the correct answer is D.

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