Consider the following problem:
Given an array of $n$ integer numbers (positive and negative), find a (contiguous) subarray for which the product $(\text{sum of the elements})\times(\text{length of the subarray})$ is maximum.
Is there an algorithm with complexity less than $O(n^2)$?
For instance, if the array is $$-2,\ 3,\ 7,\ -8,\ -15,\ \underline{1,\ 1,\ 7,\ -2},\ -3$$ then the underlined subarray would be the one we are looking for (its "score" is 7×4 = 28).
I tried everything I know (dynamic programming, greedy, divide and conquer, the general "fix one endpoint and find the other"), and asked many people, but nobody could find an answer.
