{"status": "success", "data": {"description_md": "In acute $\\triangle ABC$, let $D, E, F$ lie on $BC, CA, AB$ respectively such that $DE \\perp AC$, $EF \\perp AB$, and $FD \\perp BC$. Let $D', E', F'$ be the intersections of the circumcircle of $\\triangle DEF$ with $BC, CA$, $AB$ respectively. If $[AE'F] = 4[CD'E] = 9[BDF'] = 16$, if $[ABC]$ can be expressed as $\\frac{m}{n}$, where $m$ and $n$ are relatively prime positive integers, compute $m + n$\n\nNote: $[ABC]$ denotes the area of $ABC$.", "description_html": "<p>In acute <span class=\"katex--inline\">\\triangle ABC</span>, let <span class=\"katex--inline\">D, E, F</span> lie on <span class=\"katex--inline\">BC, CA, AB</span> respectively such that <span class=\"katex--inline\">DE \\perp AC</span>, <span class=\"katex--inline\">EF \\perp AB</span>, and <span class=\"katex--inline\">FD \\perp BC</span>. Let <span class=\"katex--inline\">D', E', F'</span> be the intersections of the circumcircle of <span class=\"katex--inline\">\\triangle DEF</span> with <span class=\"katex--inline\">BC, CA</span>, <span class=\"katex--inline\">AB</span> respectively. If <span class=\"katex--inline\">[AE'F] = 4[CD'E] = 9[BDF'] = 16</span>, if <span class=\"katex--inline\">[ABC]</span> can be expressed as <span class=\"katex--inline\">\\frac{m}{n}</span>, where <span class=\"katex--inline\">m</span> and <span class=\"katex--inline\">n</span> are relatively prime positive integers, compute <span class=\"katex--inline\">m + n</span></p>&#10;<p>Note: <span class=\"katex--inline\">[ABC]</span> denotes the area of <span class=\"katex--inline\">ABC</span>.</p>&#10;", "hints_md": "", "hints_html": "", "editorial_md": "", "editorial_html": "", "flag_hint": "", "point_value": 6, "problem_name": "TxO Math Bowl 2024 - Individuals B - Problem 14", "can_next": true, "can_prev": true, "nxt": "/problem/txo2024indivsB-p15", "prev": "/problem/txo2024indivsB-p13"}}