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The paper deals with studying the possibility of solving contact problems of the elasticity theory, in particular the punch problem using a discrete model of a continuous medium. Mixed boundary value problem of statics of elastic body is solved. Namely, the elastic equilibrium of the body is found if the displacements of some part of its surface points are given. Physically, this corresponds to the case when, using the forces, applied to the surface points, the given displacements are imparted to these points, and the surface is fixed in this form. The difference of the solved contact problems of the elasticity theory is that forces are given for some surface points, and displacements - for others. This work is based on the idea of modeling a continuous medium using the finite-size element. The rectangle, in the corners of which there are point masses, connected by elastic links was proposed as a structural element of the discrete model, replacing the rectangular element of continuous elastic medium. To make calculations for this model it is proposed to use the method of successive displacements, which gave a good account of itself in calculating beam structures. The obtained discrete models allow effectively solve contact problems of the elasticity theory, including at any values of the Poisson's ratio. [ABSTRACT FROM AUTHOR]