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With the explosive growth of wireless data traffic, the energy-efficient design in wireless networks is becoming an inevitable trend. In this paper, we study the low-complexity design issue of energy-efficient radio resource allocation in downlink OFDMA networks. Distinguished from the existing works, we focus on the tradeoff between the energy efficiency and the total transmit power. The closed-form optimal solution is obtained by solving the first-order optimality condition of the energy efficiency maximization problem. Furthermore, based on the analytical solutions, we develop a low-complexity optimal algorithm for the energy-efficient resource allocation. Simulation results show that the proposed algorithm achieves the optimal energy efficiency with significantly lower computational complexity compared with the conventional iterative methods.