{"status": "success", "data": {"description_md": "Given $f(z) = z^2-19z$, there are complex numbers $z$ with the property that $z$, $f(z)$, and $f(f(z))$ are the vertices of a right triangle in the complex plane with a right angle at $f(z)$. There are positive integers $m$ and $n$ such that one such value of $z$ is $m+\\sqrt{n}+11i$. 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": "

Given f(z) = z^2-19z, there are complex numbers z with the property that z, f(z), and f(f(z)) are the vertices of a right triangle in the complex plane with a right angle at f(z). There are positive integers m and n such that one such value of z is m+\\sqrt{n}+11i. Find m+n.

Leading zeroes must be inputted, so if your answer is 34, then input 034. Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.

", "hints_md": "", "hints_html": "", "editorial_md": "", "editorial_html": "", "flag_hint": "", "point_value": 5, "problem_name": "2019 AIME I Problem 12", "can_next": true, "can_prev": true, "nxt": "/problem/19_aime_I_p13", "prev": "/problem/19_aime_I_p11"}}