Natural environments feature mixtures of odorants of diverse quantities, qualities and complexities. Olfactory receptor neurons (ORNs) are the first layer in the sensory pathway and transmit the olfactory signal to higher regions of the brain. Yet, the response of ORNs to mixtures is strongly non-additive, and exhibits antagonistic interactions among odorants. Here, we model the processing of mixtures by mammalian ORNs, focusing on the role of inhibitory mechanisms. Theoretically predicted response curves capture experimentally determined glomerular responses imaged by a calcium indicator expressed in ORNs of live, breathing mice. Antagonism leads to an effective "normalization" of the ensemble glomerular response, which arises from a novel mechanism involving the distinct statistical properties of receptor binding and activation, without any recurrent neuronal circuitry. Normalization allows our encoding model to outperform noninteracting models in odor discrimination tasks, and to explain several psychophysical experiments in humans.