Wireless powered sensor networks (WPSNs) have emerged as a key development toward the future self-sustainable Internet of Things networks. To achieve a good balance between self-sustainability and reliability, partially WPSNs with a mixed power solution are desirable for practical applications. Specifically, most of the sensor nodes are wireless powered but the key sensor node adopts traditional wire/battery power for reliability. As a result, this paper mainly investigates optimal resource allocation for the partially WPSNs in which simultaneous wireless information and power transfer (SWIPT) is adopted in the downlink. Two scenarios with space division multiple access (SDMA) and time division multiple access (TDMA) in the uplink are considered. For both SDMA-enabled and TDMA-enabled partially WPSNs, joint design of downlink beamforming, uplink beamforming, and time allocation is investigated to maximize the uplink sum rate while guaranteeing the quality-of-service (i.e., satisfying the downlink rate constraint) at the key sensor node. After analyzing the feasibility of uplink sum rate maximization problems and the influence of the downlink rate constraint, semiclosed-form optimal solutions for both SDMA-enabled and TDMA-enabled WPSNs are proposed with guaranteed global optimality and low complexity. Extension to partially WPSNs with multiple traditionally wire/battery powered sensor nodes is also discussed for generality. The effectiveness and optimality of the proposed optimal solutions are finally demonstrated by simulations.