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Magnetic tunnel junctions can serve as ultrasensitive low-frequency magnetic sensors, however, their low-frequency performance is limited by low-frequency noise, i.e., 1/f noise. In this paper, we investigate the 1/f noise in MgO magnetic tunnel junctions (MTJs) with a tunneling magnetoresistance (TMR) of 160%, and examine the influence of annealing and MTJ size. The results show that the annealing process can not only dramatically improve the TMR, but can also strongly decrease the MTJ noise. The effect is discussed in terms of the structure of MgO barriers and tunneling probabilities. Increasing the MTJ area to 6400 μm2 yields a voltage spectral density as low as 11 nV/Hz1/2 at 1000 Hz. The possible reasons for the area dependence are discussed.