Steady-state kinetics of catalytic reactions, occurring on supported catalyst particles of the nanometer size, are analyzed by employing two schemes taking into account respectively (i) the interplay of reactions on different facets of the catalyst particle and (ii) the possibility of adsorption of reactants on the support followed by diffusion to the catalyst. A chosen model reaction, 2A + B-2 --> 2AB, is assumed to involve monomolecular A adsorption, dissociative B-2 adsorption, and the Langmuir-Hinshelwood step resulting in the formation of AB. This reaction mimics such practically important catalytic processes as CO oxidation on Pt, Rh, or Pd, and its kinetics for the infinite surface have been explored in detail during the past decade. The results obtained in the present study demonstrate that the kinetics on the nanometer-size catalyst particles can be remarkably different from those corresponding to the infinite surface. This conclusion is drawn on the basis of Monte Carlo simulations with realistic ratio between the rates of reaction and reactant diffusion [scheme (i)] and self-consistent equations explicitly describing A diffusion on the support and the reaction steps on the catalyst [scheme (ii)].