{"status": "success", "data": {"description_md": "Let $\\omega$ be a circle with radius $5$ and let $\\Gamma$ be a circle with diameter $1$ that is internally tangent to $\\omega$ at point $T$. Let $AB$ be a chord of $\\omega$ that is tangent to $\\Gamma$ at point $C$ such that $AB=8$. Given that $TC^2=\\frac{m}{n}$ for relatively prime positive integers $m$ and $n$, find $m+n$.\n\n<img src=\"https://cdn.discordapp.com/attachments/1191085670656114759/1191460283105939516/image.png\" width=\"500px\">", "description_html": "<p>Let <span class=\"katex--inline\">\\omega</span> be a circle with radius <span class=\"katex--inline\">5</span> and let <span class=\"katex--inline\">\\Gamma</span> be a circle with diameter <span class=\"katex--inline\">1</span> that is internally tangent to <span class=\"katex--inline\">\\omega</span> at point <span class=\"katex--inline\">T</span>. Let <span class=\"katex--inline\">AB</span> be a chord of <span class=\"katex--inline\">\\omega</span> that is tangent to <span class=\"katex--inline\">\\Gamma</span> at point <span class=\"katex--inline\">C</span> such that <span class=\"katex--inline\">AB=8</span>. Given that <span class=\"katex--inline\">TC^2=\\frac{m}{n}</span> for relatively prime positive integers <span class=\"katex--inline\">m</span> and <span class=\"katex--inline\">n</span>, find <span class=\"katex--inline\">m+n</span>.</p>&#10;<img src=\"https://cdn.discordapp.com/attachments/1191085670656114759/1191460283105939516/image.png\" width=\"500px\"/>&#10;", "hints_md": "", "hints_html": "", "editorial_md": "", "editorial_html": "", "flag_hint": "", "point_value": 4, "problem_name": "Christmas Contest - Team Round - Problem 16", "can_next": true, "can_prev": true, "nxt": "/problem/christmas1_team-p17", "prev": "/problem/christmas1_team-p15"}}