A piecewise-homogeneous elastic orthotropic plate, reinforced with a finite wedge-shaped inclusion, which meets the interface at a right angle and is loaded with normal forces is considered. The normal contact stresses along the contact line are determined and the behavior of the contact stresses in the neighborhood of singular points is established. By using methods of the theory of analytic functions, the problem is reduced to a singular integro-differential equation in a finite interval. Using an integral transformation a Riemann problem is obtained, the solution of which is presented in explicit form.