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A one-dimensional optimal control problem with a state-dependent cost and a unimodular integrand is considered. It is shown that, under some standard as- sumptions, this problem can be solved without using the Pontryagin maximum principle, by simple methods of the classical analysis, basing on the Tchyaplygin comparison theo- rem. However, in some modifications of the problem, the usage of Pontryagin’s maximum principle is preferable. The optimal synthesis for the problem and for its modifications is obtained.