1991 AIME Problem 3

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

{1000 \choose 0}(0.2)^0+{1000 \choose 1}(0.2)^1+{1000 \choose 2}(0.2)^2+\cdots+{1000 \choose 1000}(0.2)^{1000}

= A_0 + A_1 + A_2 + \cdots + A_{1000},

where A_k = {1000 \choose 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.