{"status": "success", "data": {"description_md": "The polynomial $$P(x)=(1+x+x^2+\\cdots+x^{17})^2-x^{17} $$has 34 complex roots of the form $z_k=r_k[\\cos(2\\pi a_k)+i\\sin(2\\pi a_k)]$, $k=1, 2, 3,\\ldots, 34$, with $00$. Given that $a_1+a_2+a_3+a_4+a_5=m/n$, 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": "

The polynomial P(x)=(1+x+x^2+\\cdots+x^{17})^2-x^{17} has 34 complex roots of the form z_k=r_k[\\cos(2\\pi a_k)+i\\sin(2\\pi a_k)], k=1, 2, 3,\\ldots, 34, with 0<a_1\\le a_2\\le a_3\\le\\cdots\\le a_{34}<1 and r_k>0. Given that a_1+a_2+a_3+a_4+a_5=m/n, where m and n are relatively prime positive integers, 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": 6, "problem_name": "2004 AIME I Problem 13", "can_next": true, "can_prev": true, "nxt": "/problem/04_aime_I_p14", "prev": "/problem/04_aime_I_p12"}}