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In this paper, we address the problem of cooperatively detecting a primary user (PU) among multiple cognitive users (CUs) when their location information is available at a CU base station. For fast detection, each CU reports a power estimate, based on one-snapshot observation of the radio environment, to the CU base station. A generalized likelihood ratio test (GLRT) is developed at the CU base station to first estimate the transmit power of the PU and then form a test variable for detection. The maximum likelihood estimator (MLE) of the unknown transmit power is discussed and analyzed to offer insight into the proposed cooperative spectrum-sensing scheme. In addition, a weighted average estimator (WAE) is proposed, which is computationally more efficient than the MLE. Asymptotic analysis for the proposed GLRT is presented. Performance of the MLE and WAE is examined along with the corresponding Cramer-Rao bound (CRB). Extensive comparisons between the proposed GLRTs and the hard- and soft-decision based spectrum sensing methods are provided, which show the effectiveness of the proposed detector.