Abstract The pleine static contact problem of elasticity theory concerning the impression of one wedge-shaped body into another of different material along sections of the side surfaces is examined. The abutting sections of both wedges start from the vertices. The problem is solved taking friction into account. In the case of greatest interest for applications, when the aperture angle of one of the wedges is π, an exact closed solution is constructed in the form of Cauchy-type integrals. However, the method of solution can be used for any wedge aperture angle.