Right off, sorry for the title, couldnt think of a good one.

Suppose I have a 2 vectors vector P = [ p_1, p_2,...p_n]'
and Q = [ q_1, q_2',...q_n'] where all of the elements in both of these vectors sum up to 1.

Prove that if a sequence of vectors  {b_k}, b_k \in R^n satisfies
 \sum_{j}^{n}b_{kj}p_j \geq \sum_{j}^{n}b_{kj}q_j for all k = 1...n, then it must be the case that  b_{kj} = A_k*b_{1j} for all j. Where A_k is just some non-negative constant.