{"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 $\\cdot$ represents multiplication). The solution to the equation $2016 \\,\\diamondsuit\\, (6\\,\\diamondsuit\\, x)=100$ can be written as $\\tfrac{p}{q}$, where $p$ and $q$ are relatively prime positive integers. What is $p+q?$\n\n$\\textbf{(A) }109\\qquad\\textbf{(B) }201\\qquad\\textbf{(C) }301\\qquad\\textbf{(D) }3049\\qquad\\textbf{(E) }33,601$", "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  <span class=\"katex--inline\">\\cdot</span>  represents multiplication). The solution to the equation  <span class=\"katex--inline\">2016 \\,\\diamondsuit\\, (6\\,\\diamondsuit\\, x)=100</span>  can be written as  <span class=\"katex--inline\">\\tfrac{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>\n<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>\n<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": 4, "problem_name": "2016 AMC 10A Problem 23", "can_next": true, "can_prev": true, "nxt": "/problem/16_amc10A_p24", "prev": "/problem/16_amc10A_p22"}}