2003 AMC 10A Problem 6

Below is the professionally curated solution for Problem 6 of the 2003 AMC 10A, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2003 AMC 10A solutions, or check the answer key.

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Concepts:custom operation absolute value counterexample

Difficulty rating: 1200

Define $x \heartsuit y$ to be $|x - y|$ for all real numbers $x$ and $y.$ Which of the following statements is not true?

$x \heartsuit y = y \heartsuit x$ for all $x$ and $y$

$2(x \heartsuit y) = (2x) \heartsuit (2y)$ for all $x$ and $y$

$x \heartsuit 0 = x$ for all $x$

$x \heartsuit x = 0$ for all $x$

$x \heartsuit y \gt 0$ if $x \ne y$

Solution:

Statement (C) claims $x \heartsuit 0 = x,$ but $x \heartsuit 0 = |x - 0| = |x|,$ which fails for negative $x.$ For example, $-1 \heartsuit 0 = 1 \ne -1.$

The remaining statements all follow directly from the properties of absolute value.

Thus, the correct answer is C.

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