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This paper investigates the problem of spectrally efficient operation of a multiuser uplink cognitive radio system in the presence of a single primary link. The secondary system applies opportunistic interference cancelation (OIC) and decodes the primary signal when such an opportunity is created. We derive the achievable rate in the secondary system when OIC is used. This scheme has a practical significance, since it enables rate adaptation without requiring any action from the primary system. The exact expressions for outage probability of the primary user are derived, when the primary system is exposed to interference from secondary users. Moreover, approximated formulas and tight lower and upper bounds for the ergodic sum-rate capacity of the secondary network are found. Next, the power allocation is investigated in the secondary system for maximizing the sum-rate under an outage constraint at the primary system. We formulate the power optimization problem in various scenarios depending on the availability of channel state information and the type of power constraints, and propose a set of simple solutions. Finally, the analytical results are confirmed by simulations, indicating both the accuracy of the analysis, and the fact that the spectral-efficient, low-complexity, flexible, and high-performing cognitive radio can be designed based on the proposed schemes.