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We find an exact solution for the problem of electron spin relaxation in a two-dimensional (2D) circle with Rashba spin-orbit interaction. Our analysis shows that the spin relaxation in finite-size regions involves three stages and is described by multiple spin relaxation times. It is important that the longest spin relaxation time increases with the decrease in system radius but always remains finite. Therefore, at long times, the spin polarization in small 2D systems decays exponentially with a size-dependent rate. This prediction is supported by results of Monte Carlo simulations.

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