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Multi-access edge computing (MEC) can enhance the computing capability of mobile devices, while non-orthogonal multiple access (NOMA) can provide high data rates. Combining these two strategies can effectively benefit the network with spectrum and energy efficiency. In this paper, we investigate the task delay minimization in multi-user NOMA-MEC networks, where multiple users can offload their tasks simultaneously through the same frequency band. We adopt the partial offloading policy, in which each user can partition its computation task into offloading and locally computing parts. We aim to minimize the task delay among users by optimizing their tasks partition ratios and offloading transmit power. The delay minimization problem is first formulated, and it is shown that it is a nonconvex one. By carefully investigating its structure, we transform the original problem into an equivalent quasi-convex. In this way, a bisection search iterative algorithm is proposed in order to achieve the minimum task delay. To reduce the complexity of the proposed algorithm and evaluate its optimality, we further derive closed-form expressions for the optimal task partition ratio and offloading power for the case of two-user NOMA-MEC networks. Simulations demonstrate the convergence and optimality of the proposed algorithm and the effectiveness of the closed-form analysis.