Combinatorial sensor arrays, such as the olfactory system, can detect a large number of analytes using a relatively small number of receptors. However, the complex pattern of receptor responses to even a single analyte, coupled with the non-linearity of responses to mixtures of analytes, makes quantitative prediction of compound concentrations in a mixture a challenging task. Here we develop a physical model that explicitly takes receptor-ligand interactions into account, and apply it to infer concentrations of highly related sugar nucleotides from the output of four engineered G-protein-coupled receptors. We also derive design principles that enable accurate mixture discrimination with cross-specific sensor arrays. The optimal sensor parameters exhibit relatively weak dependence on component concentrations, making a single designed array useful for analyzing a sizable range of mixtures. The maximum number of mixture components that can be successfully discriminated is twice the number of sensors in the array. Finally, antagonistic receptor responses, well-known to play an important role in natural olfactory systems, prove to be essential for the accurate prediction of component concentrations.