<p>This article discusses two axisymmetric contact problems for a piecewise homogeneous space. Space consists of two dissimilar half-spaces with an interphase disc-shaped crack, onto one of the edges of which an absolutely rigid stamp is pressed, taking into account static friction. The radius of stamp is less than or equal to the radius of the crack. By using discontinuous solutions of the axisymmetric theory of elasticity, the governing equation of the problems is obtained both in the form of one singular integral equation of the second kind with respect to the reduced unknown contact pressure in the first case and a system of singular integral equations with respect to reduced normal contact pressure and dislocations of the displacements points of the crack edges in the second case. The solution of the governing equations in both cases is constructed using the numerical-analytical method of mechanical quadratures. A numerical calculation was carried out and patterns of change of important physical and mechanical characteristics for problems were identified depending on the physical, mechanical and geometric characteristics.</p>