2003 AMC 10A Problems/Problem 6

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Problem

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

$\mathrm{(A) \ } x \heartsuit y = y \heartsuit x$ for all $x$ and $y$

$\mathrm{(B) \ } 2(x \heartsuit y) = (2x) \heartsuit (2y)$ for all $x$ and $y$

$\mathrm{(C) \ } x \heartsuit 0 = x$ for all $x$

$\mathrm{(D) \ } x \heartsuit x = 0$ for all $x$

$\mathrm{(E) \ } x \heartsuit y > 0$ if $x \neq y$

Solution

Examining statement C:

$x \heartsuit 0 = |x-0| = |x|$

$|x| \neq x$ when $x<0$ , but statement C says that it does for all $x$ .

Therefore the statement that is not true is " $x \heartsuit 0 = x$ for all $x$ " $\Rightarrow C$

Alternatively, consider that the given "heart function" is actually the definition of the distance between two points. Examining all of the statements, only C is not necessarily true; if c is negative, the distance between $c$ and $0$ is the absolute value of $c$ , not $c$ itself, because distance is always nonnegative.

2003 AMC 10A (Problems • Resources)
Preceded by Problem 5	Followed by Problem 7
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2003 AMC 10A Problems/Problem 6

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