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(a) Consider an ellipse in the plane. Prove that there exists a Riemannian metric which is defined on the whole plane, and with respect to which the ellipse is a geodesic. Prove that the Gaussian curvature of any such Riemannian metric takes a positive value. (b) Consider two nonintersecting, simple closed smooth curves in the plane. Prove that if there is a Riemmanian metric defined on the whole plane and the two curves are geodesics of that metric, then the Gaussian curvature of the metric vanishes somewhere.
Proposed by Tran Quoc Binh

Problem Statement