{"status": "success", "data": {"description_md": "Let $a_1, a_2, \\ldots a_n$ be a sequence defined such that $a_1 = \\tfrac{1}{6}$ and $a_{n-1} = a_{n-2} + \\frac{1}{n^3-n}$. For how many of the first $100$ terms of the sequence is the numerator of $a_i$ even when it is fully simplified?", "description_html": "<p>Let <span class=\"katex--inline\">a_1, a_2, \\ldots a_n</span> be a sequence defined such that <span class=\"katex--inline\">a_1 = \\tfrac{1}{6}</span> and <span class=\"katex--inline\">a_{n-1} = a_{n-2} + \\frac{1}{n^3-n}</span>. For how many of the first <span class=\"katex--inline\">100</span> terms of the sequence is the numerator of <span class=\"katex--inline\">a_i</span> even when it is fully simplified?</p>&#10;", "hints_md": "", "hints_html": "", "editorial_md": "", "editorial_html": "", "flag_hint": "", "point_value": 4, "problem_name": "Holiday Contest 2025 - Team Round - Problem 16", "can_next": false, "can_prev": false, "nxt": "", "prev": ""}}