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Cognitive radio (CR) is a promising concept for improving the utilization of scarce radio spectrum resources. Orthogonal frequency division multiplexing (OFDM) is regarded as a technology which is well-matched for CR systems. Dynamic resource allocation is an important task in such systems. In this paper, novel multiuser resource allocation algorithms for OFDM CR systems are presented. The algorithms attempt to maximize the total transmit bit rate (system throughput) of a group of secondary (unlicensed or CR) users subject to (1) a total transmit power constraint for secondary users, (2) a maximum tolerable interference level which can be tolerated by primary (licensed) users. The algorithms are fair in the sense that they try to allocate bits to users who have not received their fair share of service as much as possible. Simulation results show that the proposed algorithm achieves a performance close to optimal. The effect on system throughput of changing various system parameter values is also examined.