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Methods of solving an integral equation that represents the elastic contact of a strip pressed between two identical cylinders are discussed. Fourier cosine transformation is used to derive the integral equation of the contact problem, and approximation kernels are used to obtain the solutions for the cases of thick and thin layers. The solutions for both cases are given as a truncated series of the weighted Tchebyshev polynomials of the first kind, whose coefficients are determined from a system of linear equations. The problem of two circular cylinders pressing on a strip is given as an example. The nature of two approximate solutions are also briefly discussed.