{"status": "success", "data": {"description_md": "Define $x \\heartsuit y$ to be $|x-y|$ for all real numbers $x$ and $y$. Which of the following statements is not true? \n\n$\\mathrm{(A) \\ } x \\heartsuit y = y \\heartsuit x$ for all $x$ and $y$\n\n$\\mathrm{(B) \\ } 2(x \\heartsuit y) = (2x) \\heartsuit (2y)$ for all $x$ and $y$ \n\n$\\mathrm{(C) \\ } x \\heartsuit 0 = x$ for all $x$ \n\n$\\mathrm{(D) \\ } x \\heartsuit x = 0$ for all $x$ \n\n$\\mathrm{(E) \\ } x \\heartsuit y > 0$ if $x \\neq y$\n___\nFull credit goes to [MAA](https://maa.org/) for authoring these problems. These problems were taken on the [AOPS](https://artofproblemsolving.com/) website.", "description_html": "<p>Define  <span class=\"katex--inline\">x \\heartsuit y</span>  to be  <span class=\"katex--inline\">|x-y|</span>  for all real numbers  <span class=\"katex--inline\">x</span>  and  <span class=\"katex--inline\">y</span> . Which of the following statements is not true?</p>&#10;<p> <span class=\"katex--inline\">\\mathrm{(A) \\ } x \\heartsuit y = y \\heartsuit x</span>  for all  <span class=\"katex--inline\">x</span>  and  <span class=\"katex--inline\">y</span> </p>&#10;<p> <span class=\"katex--inline\">\\mathrm{(B) \\ } 2(x \\heartsuit y) = (2x) \\heartsuit (2y)</span>  for all  <span class=\"katex--inline\">x</span>  and  <span class=\"katex--inline\">y</span> </p>&#10;<p> <span class=\"katex--inline\">\\mathrm{(C) \\ } x \\heartsuit 0 = x</span>  for all  <span class=\"katex--inline\">x</span> </p>&#10;<p> <span class=\"katex--inline\">\\mathrm{(D) \\ } x \\heartsuit x = 0</span>  for all  <span class=\"katex--inline\">x</span> </p>&#10;<p> <span class=\"katex--inline\">\\mathrm{(E) \\ } x \\heartsuit y &gt; 0</span>  if  <span class=\"katex--inline\">x \\neq y</span> </p>&#10;<hr><p>Full credit goes to <a href=\"https://maa.org/\">MAA</a> for authoring these problems. These problems were taken on the <a href=\"https://artofproblemsolving.com/\">AOPS</a> website.</p>", "hints_md": "", "hints_html": "", "editorial_md": "", "editorial_html": "", "flag_hint": "", "point_value": 2, "problem_name": "2003 AMC 12A Problem 6", "can_next": true, "can_prev": true, "nxt": "/problem/03_amc12A_p07", "prev": "/problem/03_amc12A_p05"}}