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For an egg-shaped nanoparticle model consisting of an antiferromagnetic (AFM) core and a ferromagnetic shell, Monte Carlo methods with a modified Metropolis algorithm are performed. We unambiguously demonstrate the dependence of exchange bias (EB) and magnetization reversal mechanism on cooling field, radius, and position of AFM core by observing the microscopic spin configurations and the microscopic spin energy distributions. The results show that the influence factors for EB and magnetization reversal mechanism are different. The effects of cooling field and radius of AFM core on EB are strong. However, the magnetization reversal mechanism is influenced strongly by the radius and position of AFM core. This work will stimulate further theoretical and experimental investigations of the relation between EB and magnetization reversal mechanism.