1991 AIME Problem 15

For positive integer n_{}^{}, define S_n^{} to be the minimum value of the sum

\sum_{k=1}^n \sqrt{(2k-1)^2+a_k^2},

where a_1,a_2,\ldots,a_n^{} are positive real numbers whose sum is 17. There is a unique positive integer n^{}_{} for which S_n^{} is also an integer. Find this 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.