{"status": "success", "data": {"description_md": "For all $x$ such that $x \\in \\mathbb{R}$ and $x < 0$, let $x - \\dfrac{1}{x} = \\sqrt{5}$.\n\nIf $\\dfrac{x^{12} - x^{10} + x^6 + x^4 - x^0 + x^{-2}}{x^{12} - x^{10} + x^8 + x^2 - x^0 + x^{-2}} = \\dfrac{m}{n}$, where $m$ and $n$ are relatively prime positive integers. Compute $m + n$.", "description_html": "<p>For all <span class=\"katex--inline\">x</span> such that <span class=\"katex--inline\">x \\in \\mathbb{R}</span> and <span class=\"katex--inline\">x &lt; 0</span>, let <span class=\"katex--inline\">x - \\dfrac{1}{x} = \\sqrt{5}</span>.</p>&#10;<p>If <span class=\"katex--inline\">\\dfrac{x^{12} - x^{10} + x^6 + x^4 - x^0 + x^{-2}}{x^{12} - x^{10} + x^8 + x^2 - x^0 + x^{-2}} = \\dfrac{m}{n}</span>, where <span class=\"katex--inline\">m</span> and <span class=\"katex--inline\">n</span> are relatively prime positive integers. 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": "Christmas Contest - Guts Round - Set 5 Problem 3", "can_next": false, "can_prev": false, "nxt": "", "prev": ""}}