{"status": "success", "data": {"description_md": "Circles $\\omega_1$ and $\\omega_2$ intersect at points $X_1$ and $X_2$. A line tangent to circles $\\omega_1$ and $\\omega_2$ at $A$ and $B$, respectively, is closer to $X_1$ than $X_2$. The line through $X_1$ parallel to $AB$ intersect $\\omega_1$ and $\\omega_2$ at $D$ and $C$, respectively, different from $X_1$. $E$ is the intersection of $DA$ and $CB$. If $EC = 12$, $CD = 18$, $DE = 10$. If the length of $X_1X_2$ can be expressed as $\\frac{a\\sqrt{b}}{c}$, where $a, b,$ and $c$ are positive integers and $\\gcd(a, c) = 1$, find $a+b+c$.", "description_html": "

Circles \\omega_1 and \\omega_2 intersect at points X_1 and X_2. A line tangent to circles \\omega_1 and \\omega_2 at A and B, respectively, is closer to X_1 than X_2. The line through X_1 parallel to AB intersect \\omega_1 and \\omega_2 at D and C, respectively, different from X_1. E is the intersection of DA and CB. If EC = 12, CD = 18, DE = 10. If the length of X_1X_2 can be expressed as \\frac{a\\sqrt{b}}{c}, where a, b, and c are positive integers and \\gcd(a, c) = 1, find a+b+c.

", "hints_md": "", "hints_html": "", "editorial_md": "", "editorial_html": "", "flag_hint": "", "point_value": 6, "problem_name": "TxO Math Bowl 2024 - Individuals A - Problem 13", "can_next": true, "can_prev": true, "nxt": "/problem/txo2024indivsA-p14", "prev": "/problem/txo2024indivsA-p12"}}