{"status": "success", "data": {"description_md": "An equilateral triangle $ABC$ with side length $\\sqrt3$ is drawn in the plane. Two congruent circles, $\\omega_1$ and $\\omega_2$ are drawn internally tangent to the circumcircle of $ABC$ at $B$ and $C$, respectively. As well, $\\omega_1$ and $\\omega_2$ are externally tangent at some point. Circle $\\omega_3$ is drawn in the interior of $ABC$ such that it is tangent to $AB,AC$ and externally tangent to $\\omega_1,$ and $\\omega_2$. The radius of $\\omega_3$ can be written as $a-\\sqrt{\\sqrt b- c}$, where $a$, $b$, and $c$ are positive integers. Find the value of $a+b+c$.", "description_html": "<p>An equilateral triangle <span class=\"katex--inline\">ABC</span> with side length <span class=\"katex--inline\">\\sqrt3</span> is drawn in the plane. Two congruent circles, <span class=\"katex--inline\">\\omega_1</span> and <span class=\"katex--inline\">\\omega_2</span> are drawn internally tangent to the circumcircle of <span class=\"katex--inline\">ABC</span> at <span class=\"katex--inline\">B</span> and <span class=\"katex--inline\">C</span>, respectively. As well, <span class=\"katex--inline\">\\omega_1</span> and <span class=\"katex--inline\">\\omega_2</span> are externally tangent at some point. Circle <span class=\"katex--inline\">\\omega_3</span> is drawn in the interior of <span class=\"katex--inline\">ABC</span> such that it is tangent to <span class=\"katex--inline\">AB,AC</span> and externally tangent to <span class=\"katex--inline\">\\omega_1,</span> and <span class=\"katex--inline\">\\omega_2</span>. The radius of <span class=\"katex--inline\">\\omega_3</span> can be written as <span class=\"katex--inline\">a-\\sqrt{\\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. Find the value of <span class=\"katex--inline\">a+b+c</span>.</p>&#10;", "hints_md": "", "hints_html": "", "editorial_md": "", "editorial_html": "", "flag_hint": "", "point_value": 6, "problem_name": "TxO Math Bowl 2024 - Guts Contest - Set 7 Problem 1", "can_next": false, "can_prev": false, "nxt": "", "prev": ""}}