{"status": "success", "data": {"description_md": "Let
\n$$P(x)=24x^{24}+\\sum_{j=1}^{23}(24-j)(x^{24-j}+x^{24+j}). $$Let $z_{1},z_{2},\\ldots,z_{r}$ be the distinct zeros of $P(x),$ and let $z_{k}^{2}=a_{k}+b_{k}i$ for $k=1,2,\\ldots,r,$ where $i=\\sqrt{-1},$ and $a_{k}$ and $b_{k}$ are real numbers. Let
\n$$\\sum_{k=1}^{r}|b_{k}|=m+n\\sqrt{p}, $$where $m,$ $n,$ and $p$ are integers and $p$ is not divisible by the square of any prime. Find $m+n+p.$\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": "

Let
P(x)=24x^{24}+\\sum_{j=1}^{23}(24-j)(x^{24-j}+x^{24+j}). Let z_{1},z_{2},\\ldots,z_{r} be the distinct zeros of P(x), and let z_{k}^{2}=a_{k}+b_{k}i for k=1,2,\\ldots,r, where i=\\sqrt{-1}, and a_{k} and b_{k} are real numbers. Let
\\sum_{k=1}^{r}|b_{k}|=m+n\\sqrt{p}, where m, n, and p are integers and p is not divisible by the square of any prime. Find m+n+p.

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": 6, "problem_name": "2003 AIME II Problem 15", "can_next": false, "can_prev": true, "nxt": "", "prev": "/problem/03_aime_II_p14"}}