{"status": "success", "data": {"description_md": "A function $f$ is defined on the complex numbers by $f(z)=(a+bi)z,$ where $a$ and $b$ are positive numbers. This function has the property that the image of each point in the complex plane is equidistant from that point and the origin. Given that $|a+bi|=8$ and that $b^2=m/n,$ where $m$ and $n$ are relatively prime positive integers. Find $m+n.$\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>A function <span class=\"katex--inline\">f</span> is defined on the complex numbers by <span class=\"katex--inline\">f(z)=(a+bi)z,</span> where <span class=\"katex--inline\">a</span> and <span class=\"katex--inline\">b</span> are positive numbers. This function has the property that the image of each point in the complex plane is equidistant from that point and the origin. Given that <span class=\"katex--inline\">|a+bi|=8</span> and that <span class=\"katex--inline\">b^2=m/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;<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": "1999 AIME Problem 9", "can_next": true, "can_prev": true, "nxt": "/problem/99_aime_p10", "prev": "/problem/99_aime_p08"}}