{"status": "success", "data": {"description_md": "$\\triangle ABC$ satisfies $AB=AC$, and has circumcircle $\\omega$. A point $P$ lies on the arc $BC$ not containing $A$ such that $PB<PC$. Let $Q$ be the reflection of $P$ over point $B$, and suppose that $\\overline{AQ} \\perp \\overline{BQ}$. Given that $AB=7$ and $AQ=5$, the area of $\\triangle ABC$ can be expressed as $\\tfrac{a\\sqrt{b}}{c}$, where $a$, $b$, and $c$ are positive integers such that $a$ and $c$ are relatively prime, and $b$ is not divisible by the square of any prime, find $a+b+c$.", "description_html": "<p><span class=\"katex--inline\">\\triangle ABC</span> satisfies <span class=\"katex--inline\">AB=AC</span>, and has circumcircle <span class=\"katex--inline\">\\omega</span>. A point <span class=\"katex--inline\">P</span> lies on the arc <span class=\"katex--inline\">BC</span> not containing <span class=\"katex--inline\">A</span> such that <span class=\"katex--inline\">PB&lt;PC</span>. Let <span class=\"katex--inline\">Q</span> be the reflection of <span class=\"katex--inline\">P</span> over point <span class=\"katex--inline\">B</span>, and suppose that <span class=\"katex--inline\">\\overline{AQ} \\perp \\overline{BQ}</span>. Given that <span class=\"katex--inline\">AB=7</span> and <span class=\"katex--inline\">AQ=5</span>, the area of <span class=\"katex--inline\">\\triangle ABC</span> can be expressed as <span class=\"katex--inline\">\\tfrac{a\\sqrt{b}}{c}</span>, where <span class=\"katex--inline\">a</span>, <span class=\"katex--inline\">b</span>, and <span class=\"katex--inline\">c</span> are positive integers such that <span class=\"katex--inline\">a</span> and <span class=\"katex--inline\">c</span> are relatively prime, and <span class=\"katex--inline\">b</span> is not divisible by the square of any prime, find <span class=\"katex--inline\">a+b+c</span>.</p>&#10;", "hints_md": "", "hints_html": "", "editorial_md": "", "editorial_html": "", "flag_hint": "", "point_value": 5, "problem_name": "TxO Math Bowl 2024 - Individuals A - Problem 11", "can_next": true, "can_prev": true, "nxt": "/problem/txo2024indivsA-p12", "prev": "/problem/txo2024indivsA-p10"}}