# 2003 AMC 12A Problems/Problem 6

The following problem is from both the 2003 AMC 12A #6 and 2003 AMC 10A #6, so both problems redirect to this page.

## 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

We start by looking at the answers. Examining statement C, we notice: $x \heartsuit 0 = |x-0| = |x|$ $|x| \neq x$ when $x<0$, but statement C says that it does for all $x$.

Therefore the statement that is not true is $\boxed{\mathrm{(C)}\ x\heartsuit 0=x\ \text{for all}\ x}$

## Video Solution

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 