{"status": "success", "data": {"description_md": "Triangle $ABC$ is constructed such that $\\angle A = 60^\\circ$, $AC=8\\sqrt{7}$, and $BC=25$. Denoting the incenter as $I$ and orthocenter as $H$, the value of $\\sin (\\angle AHI)$ can be expressed as $\\tfrac{m}{n}$ for relatively prime, positive integers $m$ and $n$. Given that $AB>AC$, find $m+n$.\n\n$\\textbf{(A)}~7\\qquad\\textbf{(B)}~169\\qquad\\textbf{(C)}~196\\qquad\\textbf{(D)}~242\\qquad\\textbf{(E)}~243$", "description_html": "<p>Triangle <span class=\"katex--inline\">ABC</span> is constructed such that <span class=\"katex--inline\">\\angle A = 60^\\circ</span>, <span class=\"katex--inline\">AC=8\\sqrt{7}</span>, and <span class=\"katex--inline\">BC=25</span>. Denoting the incenter as <span class=\"katex--inline\">I</span> and orthocenter as <span class=\"katex--inline\">H</span>, the value of <span class=\"katex--inline\">\\sin (\\angle AHI)</span> can be expressed as <span class=\"katex--inline\">\\tfrac{m}{n}</span> for relatively prime, positive integers <span class=\"katex--inline\">m</span> and <span class=\"katex--inline\">n</span>. Given that <span class=\"katex--inline\">AB&gt;AC</span>, find <span class=\"katex--inline\">m+n</span>.</p>&#10;<p><span class=\"katex--inline\">\\textbf{(A)}~7\\qquad\\textbf{(B)}~169\\qquad\\textbf{(C)}~196\\qquad\\textbf{(D)}~242\\qquad\\textbf{(E)}~243</span></p>&#10;", "hints_md": "", "hints_html": "", "editorial_md": "", "editorial_html": "", "flag_hint": "", "point_value": 5, "problem_name": "2024 Mock AMC 12 - Problem 25", "can_next": false, "can_prev": true, "nxt": "", "prev": "/problem/2024_mock_amc12-p24"}}