{"status": "success", "data": {"description_md": "A binary operation $\\diamondsuit$ has the properties that $a\\ \\diamondsuit\\ (b\\ \\diamondsuit\\ c) = (a\\ \\diamondsuit\\ b)\\cdot c$ and that $a\\ \\diamondsuit\\ a = 1$ for all nonzero real numbers $a, b$ and $c.$ (Here the dot  $\\cdot$  represents the usual multiplication operation.) The solution to the equation $2016\\ \\diamondsuit\\ (6\\ \\diamondsuit\\ x) = 100$ can be written as $\\frac{p}{q},$ where $p$ and $q$ are relatively prime positive integers. What is $p + q?$ \n\n$\\textbf{(A)}\\ 109\\qquad\\textbf{(B)}\\ 201\\qquad\\textbf{(C)}\\ 301\\qquad\\textbf{(D)}\\ 3049\\qquad\\textbf{(E)}\\ 33,601$\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>A binary operation  <span class=\"katex--inline\">\\diamondsuit</span>  has the properties that  <span class=\"katex--inline\">a\\ \\diamondsuit\\ (b\\ \\diamondsuit\\ c) = (a\\ \\diamondsuit\\ b)\\cdot c</span>  and that  <span class=\"katex--inline\">a\\ \\diamondsuit\\ a = 1</span>  for all nonzero real numbers  <span class=\"katex--inline\">a, b</span>  and  <span class=\"katex--inline\">c.</span>  (Here the dot   <span class=\"katex--inline\">\\cdot</span>   represents the usual multiplication operation.) The solution to the equation  <span class=\"katex--inline\">2016\\ \\diamondsuit\\ (6\\ \\diamondsuit\\ x) = 100</span>  can be written as  <span class=\"katex--inline\">\\frac{p}{q},</span>  where  <span class=\"katex--inline\">p</span>  and  <span class=\"katex--inline\">q</span>  are relatively prime positive integers. What is  <span class=\"katex--inline\">p + q?</span> </p>&#10;<p> <span class=\"katex--inline\">\\textbf{(A)}\\ 109\\qquad\\textbf{(B)}\\ 201\\qquad\\textbf{(C)}\\ 301\\qquad\\textbf{(D)}\\ 3049\\qquad\\textbf{(E)}\\ 33,601</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": 3, "problem_name": "2016 AMC 12A Problem 20", "can_next": true, "can_prev": true, "nxt": "/problem/16_amc12A_p21", "prev": "/problem/16_amc12A_p19"}}