Designing fast algorithms that adapt the transmit and receive power and beamformers to optimize performance for different users is important in wireless MIMO downlink systems. This paper studies the max-min weighted SIR problem in the downlink, where multiple users are weighted according to priority and are subject to a total power constraint. The difficulty of this nonconvex problem is compounded by the coupling in the transmit and receive beamformers, thereby making it hard to optimize in a distributed fashion. We first show that this problem can be optimally and efficiently computed using a fast algorithm when the channels are rank-one. The optimal transmit and receive power and beamformers are also derived analytically. We then exploit the MIMO uplink-downlink duality to adapt our algorithm to compute a local optimal solution for channels with general rank.