{"status": "success", "data": {"description_md": "Expanding $(1+0.2)^{1000}$ by the binomial theorem and doing no further manipulation gives\n\n$$\n\\begin{aligned} &\\ & \\binom{1000}{0}(0.2)^0+\\binom{1000}{1}(0.2)^1+\\binom{1000}{2}(0.2)^2+\\cdots+\\binom{1000}{1000}(0.2)^{1000}\\\\ &\\ & = A_0 + A_1 + A_2 + \\cdots + A_{1000}, \\end{aligned}\n$$ where $A_k = \\binom{1000}{k}(0.2)^k$ for $k = 0,1,2,\\ldots,1000$. For which $k$ is $A_k$ the largest?\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": "

Expanding (1+0.2)^{1000} by the binomial theorem and doing no further manipulation gives

\\begin{aligned} &\\ & \\binom{1000}{0}(0.2)^0+\\binom{1000}{1}(0.2)^1+\\binom{1000}{2}(0.2)^2+\\cdots+\\binom{1000}{1000}(0.2)^{1000}\\\\ &\\ & = A_0 + A_1 + A_2 + \\cdots + A_{1000}, \\end{aligned} where A_k = \\binom{1000}{k}(0.2)^k for k = 0,1,2,\\ldots,1000. For which k is A_k the largest?

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": 3, "problem_name": "1991 AIME Problem 3", "can_next": true, "can_prev": true, "nxt": "/problem/91_aime_p04", "prev": "/problem/91_aime_p02"}}