{"status": "success", "data": {"description_md": "Two circles centered at $O_1$ and $O_2$ intersect at $A$ and $B$, and a common external tangent closer to $A$ intersects the two circles at $C$, $D$, where $C$ is on the circle with center $O_1$. The extension of segment $BA$  intersects $CD$ at $E$. If $O_1O_2 = 10$, $O_1C=6$, and $O_2D=14$, then $AE$ can be written as $a-b\\sqrt{c}$, where $a$, $b$, and $c$ are positive integers with $c$ not divisible by the square of any prime. Find $a+b+c$.\n", "description_html": "<p>Two circles centered at <span class=\"katex--inline\">O_1</span> and <span class=\"katex--inline\">O_2</span> intersect at <span class=\"katex--inline\">A</span> and <span class=\"katex--inline\">B</span>, and a common external tangent closer to <span class=\"katex--inline\">A</span> intersects the two circles at <span class=\"katex--inline\">C</span>, <span class=\"katex--inline\">D</span>, where <span class=\"katex--inline\">C</span> is on the circle with center <span class=\"katex--inline\">O_1</span>. The extension of segment <span class=\"katex--inline\">BA</span>  intersects <span class=\"katex--inline\">CD</span> at <span class=\"katex--inline\">E</span>. If <span class=\"katex--inline\">O_1O_2 = 10</span>, <span class=\"katex--inline\">O_1C=6</span>, and <span class=\"katex--inline\">O_2D=14</span>, then <span class=\"katex--inline\">AE</span> can be written as <span class=\"katex--inline\">a-b\\sqrt{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 with <span class=\"katex--inline\">c</span> 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 - Guts Contest - Set 5 Problem 3", "can_next": false, "can_prev": false, "nxt": "", "prev": ""}}