The nonstationary indentation of a rigid blunt indenter into an elastic layer is studied. An approach to solving a mixed initial-boundary-value problem with an unknown moving boundary is developed. The problem is reduced to an infinite system of integral equations and the equation of motion of the indenter. The system is solved numerically. The analytical solution of the nonmixed problem is found for the initial stage of the indentation process Keywords: impact, elastic layer, wave diffraction, mixed problem Introduction. The nonstationary contact problem of elasticity has intensively been studied for the past two to three decades owing to its practical importance, specific features of the physical process, and interesting ways of solving the associated boundary-value problems. The current state of the art in the field is reviewed in, e.g., [1,5,11]. Methods for solving such problems are outlined in, e.g., [7,8,10,12,17]. The problem of impact of a body against an elastic medium or a structural member is generally formulated as a nonstationary mixed initial-boundary-value problem of elasticity with an unknown time-dependent boundary, which is found in the course of solution. The problem statement includes the equations of elastic deformation of the impacted body, the equation of motion of the impactor, the relationship between the interaction force and the unknown contact boundary, the relationship between the contact area and the displacement (penetration) of the impactor, and the boundary and initial conditions. In the general case, the problem is coupled one with fuzzy input data, which predetermines the difficulties of its solution.Note the following physical factor that has a direct influence on the ways of solving the problem of impact of a body against an elastic medium. During an impact, the contact boundary between the bodies moves over their surfaces with variable velocity dependent on their shapes. If the bodies are blunted, the velocity of the contact boundary can be very high during the initial period (at least, higher than the velocity of elastic waves in the body (supersonic)). During this period, waves in the elastic body do not interact with its free surface and, hence, the boundary conditions on it can be chosen as one sees fit. This would allow us, at least for the early stage of interaction, to formulate a nonmixed boundary-value problem and, thus, to simplify the solution procedure. Eventually, the velocity of the contact boundary reduces down to transonic and then subsonic magnitudes because of the geometry of the bodies and the deceleration of the impactor, which causes the waves to reach (and interact with) the surface of the body outside the contact region. However, as shown in, e.g., [9,11,13], the supersonic solution is in some cases enough to make assessments.As follows from the cited publications, most studies on the topic address a rigid or elastic body colliding with an elastic half-space. There are much fewer studies on the interaction of a blunt impactor with an elastic body of fi...