I have been thinking about it for several days and still have not solved it completely. This is the problem:
My formula: Outcome=$(a_1*b_1)+(a_2*b_2)+(a_3*b_3)...+(a_n*b_n),$ where $0\le a\le 1,$ $0\le b\le1, ~~a+b\le 1, ~~a_1+a_2+a_3...+a_n\le 1.$
I would like to find the maximum of this function based on the value of n. Just by trying out different numbers, I have found that the maximum should be (n-1)/n, but I have not been able to prove it mathematically yet. Any help is very much appreciated!