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We develop adaptive scheduling and power control algorithms for random access in a multiple access channel where terminals acquire instantaneous channel state information but do not know the probability distribution of the channel. In each time slot, terminals measure the channel to the common access point. Based on the observed channel value, they determine whether to transmit or not and, if they decide to do so, adjust their transmitted power. We remark that there is no coordination between terminals and that adaptation is to the local channel value only. It is shown that the proposed algorithm almost surely maximizes a proportional fair utility while adhering to instantaneous and average power constraints. Important properties of the algorithm are adaptivity, low computational complexity and the ability to handle non-convex rate functions. Numerical results on a randomly generated network with heterogeneous users corroborate theoretical results.