This work is devoted to the formulation and construction of an analytical solution to the problem of contact between a cylindrical viscoelastic aging pipe with an internal thin coating and an insert having a complex shape placed inside the pipe with an interference fit. In practice, the presence of such coatings is required, for example, to protect the main structure from aggressive external or internal environments, for its electrical insulation, etc. The manufacturing process of the inner coating determines its possible heterogeneity (dependence of properties on coordinates). An insert placed inside a pipe can have a complex profile that has a rapidly changing function. Taking these features into account is important when analyzing the stress-strain state of pipes with an internal coating. Using an approach based on the use of special basis functions and the type of solution, a representation for the contact stresses in the pipe in the region of the rigid insert is obtained. This approach makes it possible to distinguish functions that describe the properties of the inner coating and the shape of the outer profile of the insert in the form of separate terms and factors in the expression for the contact stresses in the insert region. Therefore, in order to achieve high accuracy when carrying out calculations, it is sufficient to restrict ourselves to a relatively small number of terms
