{"status": "success", "data": {"description_md": "Let $S$ be the set $\\{1,2,3,...,19\\}$. For $a,b \\in S$, define $a \\succ b$ to mean that either $0 < a - b \\le 9$ or $b - a > 9$. How many ordered triples $(x,y,z)$ of elements of $S$ have the property that $x \\succ y$, $y \\succ z$, and $z \\succ x$?\n\n$\\textbf{(A)} \\ 810 \\qquad  \\textbf{(B)} \\ 855 \\qquad  \\textbf{(C)} \\ 900 \\qquad  \\textbf{(D)} \\ 950 \\qquad  \\textbf{(E)} \\ 988$\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>Let  <span class=\"katex--inline\">S</span>  be the set  <span class=\"katex--inline\">\\{1,2,3,...,19\\}</span> . For  <span class=\"katex--inline\">a,b \\in S</span> , define  <span class=\"katex--inline\">a \\succ b</span>  to mean that either  <span class=\"katex--inline\">0 &lt; a - b \\le 9</span>  or  <span class=\"katex--inline\">b - a &gt; 9</span> . How many ordered triples  <span class=\"katex--inline\">(x,y,z)</span>  of elements of  <span class=\"katex--inline\">S</span>  have the property that  <span class=\"katex--inline\">x \\succ y</span> ,  <span class=\"katex--inline\">y \\succ z</span> , and  <span class=\"katex--inline\">z \\succ x</span> ?</p>&#10;<p> <span class=\"katex--inline\">\\textbf{(A)} \\ 810 \\qquad  \\textbf{(B)} \\ 855 \\qquad  \\textbf{(C)} \\ 900 \\qquad  \\textbf{(D)} \\ 950 \\qquad  \\textbf{(E)} \\ 988</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": "2013 AMC 12A Problem 20", "can_next": true, "can_prev": true, "nxt": "/problem/13_amc12A_p21", "prev": "/problem/13_amc12A_p19"}}