In this study, the frictionless contact problem between a coated layer and a half-plane is considered when they are pressed by a rigid cylindrical stamp. Upon loading, two unknown contact widths and contact pressures occur on the contact areas the advanced contact area between the stamp and the coating; and the receding contact area between the layer and the half-plane. This problem is reduced to two singular integral equations by using the Fourier transform and applying the boundary conditions of the problem. The numerical solution of the system of singular integral equations is obtained by applying the Gauss-Chebyshev integration formulas. Copyright © 2013 by IPPT PAN.