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We consider several well known nonclairvoyant scheduling problems, including the problem of minimizing the average response time, and best-effort firm real-time scheduling. It is known that there are no deterministic online algorithms for these problems with bounded (or even polylogarithmic in the number of jobs) competitive ratios. We show that moderately increasing the speed of the processor used by the non-clairvoyant scheduler effectively gives this scheduler the power of clairvoyence. Furthermore, we show that there exist online algorithms with bounded competitive ratios on all inputs that are not closely correlated with processor speed.