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

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

Prove that if a sequence of vectors $\displaystyle {b_k}, b_k \in R^n $ satisfies
$\displaystyle \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 $\displaystyle b_{kj} = A_k*b_{1j} $ for all j. Where A_k is just some non-negative constant.