{"status": "success", "data": {"description_md": "Trapezoid $ABCD$ has $\\overline{AB} \\parallel \\overline{CD}$, $BC = CD = 43$, and $\\overline{AD} \\perp \\overline{BD}$. Let $O$ be the intersection of the diagonals $\\overline{AC}$ and $\\overline{BD}$, and let $P$ be the midpoint of $\\overline{BD}$. Given that $OP = 11$, the length $AD$ can be written in the form $m\\sqrt{n}$, where $m$ and $n$ are positive integers and $n$ is not divisible by the square of any prime. What is $m + n$?\n\n$\\textbf{(A)} ~65\\qquad\\textbf{(B)} ~132\\qquad\\textbf{(C)} ~157\\qquad\\textbf{(D)} ~194\\qquad\\textbf{(E)} ~215$", "description_html": "<p>Trapezoid  <span class=\"katex--inline\">ABCD</span>  has  <span class=\"katex--inline\">\\overline{AB} \\parallel \\overline{CD}</span> ,  <span class=\"katex--inline\">BC = CD = 43</span> , and  <span class=\"katex--inline\">\\overline{AD} \\perp \\overline{BD}</span> . Let  <span class=\"katex--inline\">O</span>  be the intersection of the diagonals  <span class=\"katex--inline\">\\overline{AC}</span>  and  <span class=\"katex--inline\">\\overline{BD}</span> , and let  <span class=\"katex--inline\">P</span>  be the midpoint of  <span class=\"katex--inline\">\\overline{BD}</span> . Given that  <span class=\"katex--inline\">OP = 11</span> , the length  <span class=\"katex--inline\">AD</span>  can be written in the form  <span class=\"katex--inline\">m\\sqrt{n}</span> , where  <span class=\"katex--inline\">m</span>  and  <span class=\"katex--inline\">n</span>  are positive integers and  <span class=\"katex--inline\">n</span>  is not divisible by the square of any prime. What is  <span class=\"katex--inline\">m + n</span> ?</p>\n<p> <span class=\"katex--inline\">\\textbf{(A)} ~65\\qquad\\textbf{(B)} ~132\\qquad\\textbf{(C)} ~157\\qquad\\textbf{(D)} ~194\\qquad\\textbf{(E)} ~215</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": 3, "problem_name": "2021 AMC 10A Problem 17", "can_next": true, "can_prev": true, "nxt": "/problem/21_amc10A_p18", "prev": "/problem/21_amc10A_p16"}}