Lattice grand canonical Monte Carlo simulations of heteropolymers adsorbing on heterogeneous surfaces are performed to help understand molecular recognition. The heteropolymers are self-avoiding walks of AB copolymers, which interact with a single heterogeneous surface composed of two different types of surface sites, type I and type 2. Interactions of type A beads with type I sites and type B beads with type 2 sites are attractive, whereas other interactions are neutral. Two polymer chain types with different statistical sequences simultaneously interact with a surface with a defined statistical pattern, enabling the study of the effects of competition for the limited number of surface sites during adsorption. We find that competition results in a significant increase in selectivity. In some cases, for example, the adsorption of alternating sequences on an alternating surface completely suppresses the adsorption of other competing sequence types. The selection rules, however, remain the same as in the noncompetitive adsorption. Surfaces with extreme patterns (i.e., completely alternating or highly patchy) exhibit the highest selectivity toward the matching sequences (completely alternating or blocky, respectively). Oil moderately patchy or moderately alternating surfaces, sequences with extreme patterns (highly block), or completely alternating), not the sequences With Matching statistics, have the highest selectivity. Random surfaces have little or no selectivity toward different sequence types.