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Cell competition is recognized to be instrumental to the dynamics and structure of the tumor-host interface in invasive cancers. In mild competition scenarios, the healthy tissue and cancer cells can coexist. When the competition is aggressive, competitive cells, the so called super-competitors, expand by killing other cells. Novel chemotherapy drugs and molecularly targeted drugs are commonly administered as part of cancer therapy. Both types of drugs are susceptible to various mechanisms of drug resistance, obstructing or preventing a successful outcome. In this paper, we develop a cancer growth model that accounts for the competition between cancer cells and healthy cells. The model incorporates resistance to both chemotherapy and targeted drugs. In both cases, the level of drug resistance is assumed to be a continuous variable ranging from fully-sensitive to fully-resistant. Using our model we demonstrate that when the competition is moderate, therapies using both drugs are more effective compared with single drug therapies. However, when cancer cells are highly competitive, targeted drugs become more effective. The results of the study stress the importance of adjusting the therapy to the pre-treatment resistance levels. We conclude with a study of the spatiotemporal propagation of drug resistance in a competitive setting, verifying that the same conclusions hold in the spatially heterogeneous case.