{"status": "success", "data": {"description_md": "Given a nonnegative real number $x,$ let $\\langle x\\rangle$ denote the fractional part of $x;$ that is, $\\langle x\\rangle=x-\\lfloor x\\rfloor,$ where $\\lfloor x\\rfloor$ denotes the greatest integer less than or equal to $x.$ Suppose that $a$ is positive, $\\langle a^{-1}\\rangle=\\langle a^2\\rangle,$ and $2<a^2<3.$ Find the value of $a^{12}-144a^{-1}.$\n___\nLeading zeroes must be inputted, so if your answer is `34`, then input `034`. Full credit goes to [MAA](https://maa.org/) for authoring these problems. These problems were taken on the [AOPS](https://artofproblemsolving.com/) website.", "description_html": "<p>Given a nonnegative real number <span class=\"katex--inline\">x,</span> let <span class=\"katex--inline\">\\langle x\\rangle</span> denote the fractional part of <span class=\"katex--inline\">x;</span> that is, <span class=\"katex--inline\">\\langle x\\rangle=x-\\lfloor x\\rfloor,</span> where <span class=\"katex--inline\">\\lfloor x\\rfloor</span> denotes the greatest integer less than or equal to <span class=\"katex--inline\">x.</span> Suppose that <span class=\"katex--inline\">a</span> is positive, <span class=\"katex--inline\">\\langle a^{-1}\\rangle=\\langle a^2\\rangle,</span> and <span class=\"katex--inline\">2&lt;a^2&lt;3.</span> Find the value of <span class=\"katex--inline\">a^{12}-144a^{-1}.</span></p>&#10;<hr><p>Leading zeroes must be inputted, so if your answer is <code>34</code>, then input <code>034</code>. Full credit goes to <a href=\"https://maa.org/\">MAA</a> for authoring these problems. These problems were taken on the <a href=\"https://artofproblemsolving.com/\">AOPS</a> website.</p>", "hints_md": "", "hints_html": "", "editorial_md": "", "editorial_html": "", "flag_hint": "", "point_value": 4, "problem_name": "1997 AIME Problem 9", "can_next": true, "can_prev": true, "nxt": "/problem/97_aime_p10", "prev": "/problem/97_aime_p08"}}