{"status": "success", "data": {"description_md": "For all positive integers $n$, let the $n^\\text{th}$ golden circle be the circle with diameter $(0,0)$ to $(\\tfrac{1}{2^{n-1}},0)$. If  the area of the region of points that are enclosed within an odd number of golden circles can be written as $\\frac{m\\pi}{n}$, where $m$ and $n$ are relatively prime positive integers, find $m+n$.", "description_html": "<p>For all positive integers <span class=\"katex--inline\">n</span>, let the <span class=\"katex--inline\">n^\\text{th}</span> golden circle be the circle with diameter <span class=\"katex--inline\">(0,0)</span> to <span class=\"katex--inline\">(\\tfrac{1}{2^{n-1}},0)</span>. If  the area of the region of points that are enclosed within an odd number of golden circles can be written as <span class=\"katex--inline\">\\frac{m\\pi}{n}</span>, where <span class=\"katex--inline\">m</span> and <span class=\"katex--inline\">n</span> are relatively prime positive integers, find <span class=\"katex--inline\">m+n</span>.</p>&#10;", "hints_md": "", "hints_html": "", "editorial_md": "", "editorial_html": "", "flag_hint": "", "point_value": 4, "problem_name": "TxO Math Bowl 2024 - Guts Contest - Set 4 Problem 3", "can_next": false, "can_prev": false, "nxt": "", "prev": ""}}