In this paper, we consider a mobile edge computing (MEC) network, that is wirelessly powered. Each user harvests wireless energy and follows a binary computation offloading policy, i.e., it either executes the task locally or offloads it to the MEC as a whole. For the offloading users, non-orthogonal multiple access (NOMA) is adopted for information transmission. We consider rate-adaptive computational tasks and aim at maximizing the sum computation rate of all users by jointly optimizing the individual computing mode selection (local computing or offloading), the time allocations for energy transfer and for information transmission, together with the local computing speed or the transmission power level. The major difficulty of the rate maximization problem lies in the combinatorial nature of the multiuser computing mode selection and its involved coupling with the time allocation. We also study the case where the offloading users adopt time division multiple access (TDMA) as a benchmark, and derive the optimal time sharing among the users. We show that the maximum achievable rate is the same for the TDMA and the NOMA system, and in the case of NOMA it is independent from the decoding order, which can be exploited to improve system fairness. To maximize the sum computation rate, for the mode selection we propose a greedy solution based on the wireless channel gains, combined with the optimal allocation of energy transfer time. Numerical results show that the proposed solution maximizes the computation rate in homogeneous networks, and binary offloading leads to significant gains. Moreover, NOMA increases the fairness of rate distribution among the users significantly, when compared with TDMA.