Difference between revisions of "2003 AMC 12A Problems/Problem 6"

Revision as of 10:13, 11 November 2006

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 $x \neq y$

Examining statement C:

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

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

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

Revision as of 22:42, 9 November 2006 (view source) Xantos C. Guin (talk \| contribs) (added problem and solution)	Revision as of 10:13, 11 November 2006 (view source) JBL (talk \| contribs) m (2003 AMC 12A/Problem 6 moved to 2003 AMC 12A Problems/Problem 6) Newer edit →
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