{"status": "success", "data": {"description_md": "A basketball player has a constant probability of $.4$ of making any given shot, independent of previous shots. Let $a_{n}$ be the ratio of shots made to shots attempted after $n$ shots. The probability that $a_{10}=.4$ and $a_{n}\\le .4$ for all $n$ such that $1\\le n \\le 9$ is given to be $p^{a}q^{b}r/(s^{c}),$ where $p,$ $q,$ $r,$ and $s$ are primes, and $a,$ $b,$ and $c$ are positive integers. Find $(p+q+r+s)(a+b+c).$\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": "

A basketball player has a constant probability of .4 of making any given shot, independent of previous shots. Let a_{n} be the ratio of shots made to shots attempted after n shots. The probability that a_{10}=.4 and a_{n}\\le .4 for all n such that 1\\le n \\le 9 is given to be p^{a}q^{b}r/(s^{c}), where p, q, r, and s are primes, and a, b, and c are positive integers. Find (p+q+r+s)(a+b+c).

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": "2002 AIME II Problem 12", "can_next": true, "can_prev": true, "nxt": "/problem/02_aime_II_p13", "prev": "/problem/02_aime_II_p11"}}