A plane elastic problem for an orthotropic infinite strip with mixed boundary conditions is investigated. A model of the strip has been built by using the method of integral Fourier transforms. We obtain relationships which allow us to formulate singular integral equations for the various types of boundary conditions on one of its edges. The problems in the case of both smooth stamp-strip contact and rigid stamp-strip adhesion have been considered. It is shown that the effect of anisotropy on the contact stress distribution is minor. The stress intensity factors at the stamp corners, which are the main parameters of fracture, are evaluated. The quasi-invariance of a certain combination of the stress intensity factors is confirmed.