{"status": "success", "data": {"description_md": "Given a sequence $a_0$, $a_1$, $a_2$,..., satisfying $a_0 = 0$, $a_1 = \\sqrt{5 - 2\\sqrt 5}$, and for all $n \\ge 2$,\n\n$$a_n = \\frac{a_{n-2} a_{n-1}^2 + 2a_{n-1} - a_{n-2}}{-a_{n-1}^2 + 2a_{n-2}a_{n-1} + 1}$$\n\nFind the number of indices $k$ from $1$ to $2025$ inclusive such that $a_k$ is an integer.", "description_html": "<p>Given a sequence <span class=\"katex--inline\">a_0</span>, <span class=\"katex--inline\">a_1</span>, <span class=\"katex--inline\">a_2</span>,&#8230;, satisfying <span class=\"katex--inline\">a_0 = 0</span>, <span class=\"katex--inline\">a_1 = \\sqrt{5 - 2\\sqrt 5}</span>, and for all <span class=\"katex--inline\">n \\ge 2</span>,</p>&#10;<p><span class=\"katex--display\">a_n = \\frac{a_{n-2} a_{n-1}^2 + 2a_{n-1} - a_{n-2}}{-a_{n-1}^2 + 2a_{n-2}a_{n-1} + 1}</span></p>&#10;<p>Find the number of indices <span class=\"katex--inline\">k</span> from <span class=\"katex--inline\">1</span> to <span class=\"katex--inline\">2025</span> inclusive such that <span class=\"katex--inline\">a_k</span> is an integer.</p>&#10;", "hints_md": "", "hints_html": "", "editorial_md": "", "editorial_html": "", "flag_hint": "", "point_value": 6, "problem_name": "Grinding Aces Math Exam 2025 - Problem 9", "can_next": false, "can_prev": false, "nxt": "", "prev": ""}}