The contact of a flat, simply-connected axisymmetric indenter with a layered elastic half space is examined. The problem is mathematically formulated using integral transforms to derive singular integral equations for the contact pressure. The solution of these equations is obtained by expansion in orthogonal polynomials. The solution predicts complete contact between the indenter and the surface of the layered half space only for a restricted range of the material and geometrical parameters. Outside of this range, solutions exist with multiple contact regions. A parameter space is divided into zones for single and multiple contact solutions and comparisons are made with the solutions for the analogous plane-strain problem.