The frictionless contact problem for an elastic layer supported by two elastic quarter planes is considered. It is shown that as in the layer-half-space problem, the width of the contact area is dependent on the relative distribution of the applied load and is independent of its amplitude. At the corner of the support the contact pressure has an integrable singularity with its power α (0 < α < 0.5) depending on a single bielastic constant. The problem is reduced to a singular integral equation with a generalized Cauchy kernel and with the unknown contact pressure as the density function. The calculated results include the power and the strength of the stress singularity, the width of the contact area, and the contact pressure for various geometries, external loads, and material combinations.