The method is proposed for solving the next two axisymmetric problems of elasticity: (a) about a crack laying at the interface of layers in a layered medium, (b) about a rigid punch acting on a layered medium under the conditions of full cohesion at the contact area. The method is applied to a problem of a crack at the interface of a half-space and a layer whose second edge is joined with a different elastic half-space. Using Hankel transformation the problem has been reduced to a singular integral equation with respect to an auxiliary unknown function. The singular integral equation in the general case is regularized by Karleman-Vekua method with the aid of an analytical solution for the case of a crack at the interface of two half-spaces. Thus a system of Fredholm second kind equations has been obtained. It has been solved numerically. Results of numerical solutions are given. The behavior of the solution for different relationships between the elastic constants of the materials and the geometric parameters is discussed. An interesting property inherent precisely the layered medium was observed—a region of a stable development of a crack at the boundary in a uniform field of tensile stresses exists if the layer is formed by low-compressible material much less rigid than the half-spaces joined together by them. A qualitative explanation for this effect is proposed.