{"status": "success", "data": {"description_md": "A rhombus with diagonals length $6$ and $8$ has a circle $O$ inscribed in it. Two circles $A$ and $B$ of different radii length are both internally tangent to the rhombus at two points and also externally tangent to circle $O$ at one point. The sum of the radii of circles $A$ and $B$ can be expressed as $\\frac{m}{n}$ for relatively prime positive integers $m$ and $n$. Compute $m+n$.", "description_html": "<p>A rhombus with diagonals length <span class=\"katex--inline\">6</span> and <span class=\"katex--inline\">8</span> has a circle <span class=\"katex--inline\">O</span> inscribed in it. Two circles <span class=\"katex--inline\">A</span> and <span class=\"katex--inline\">B</span> of different radii length are both internally tangent to the rhombus at two points and also externally tangent to circle <span class=\"katex--inline\">O</span> at one point. The sum of the radii of circles <span class=\"katex--inline\">A</span> and <span class=\"katex--inline\">B</span> can be expressed as <span class=\"katex--inline\">\\frac{m}{n}</span> for relatively prime positive integers <span class=\"katex--inline\">m</span> and <span class=\"katex--inline\">n</span>. Compute <span class=\"katex--inline\">m+n</span>.</p>&#10;", "hints_md": "", "hints_html": "", "editorial_md": "", "editorial_html": "", "flag_hint": "", "point_value": 5, "problem_name": "Holiday Contest 2025 - Guts Round - Set 7 Problem 1", "can_next": false, "can_prev": false, "nxt": "", "prev": ""}}