{"status": "success", "data": {"description_md": "Consider a circle with radius $2$ and center $O$. The center of a second circle $P$ with radius $1$ is picked inside the first circle at random. Given that the two circles intersect at $2$ points, the probability that the distance between the two points of intersection is more than  than $1$ can be expressed as $\\frac{\\sqrt{a}-b}{c}$, where $a$, $b$, and $c$ are positive integers with $b$ and $c$ relatively prime. Find $a+b+c$.", "description_html": "<p>Consider a circle with radius <span class=\"katex--inline\">2</span> and center <span class=\"katex--inline\">O</span>. The center of a second circle <span class=\"katex--inline\">P</span> with radius <span class=\"katex--inline\">1</span> is picked inside the first circle at random. Given that the two circles intersect at <span class=\"katex--inline\">2</span> points, the probability that the distance between the two points of intersection is more than  than <span class=\"katex--inline\">1</span> can be expressed as <span class=\"katex--inline\">\\frac{\\sqrt{a}-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 with <span class=\"katex--inline\">b</span> and <span class=\"katex--inline\">c</span> relatively prime. Find <span class=\"katex--inline\">a+b+c</span>.</p>&#10;", "hints_md": "", "hints_html": "", "editorial_md": "", "editorial_html": "", "flag_hint": "", "point_value": 3, "problem_name": "TxO Math Bowl 2024 - Team Contest - Problem 5", "can_next": true, "can_prev": true, "nxt": "/problem/txo2024team-p06", "prev": "/problem/txo2024team-p04"}}