  We prove the existence of closed convex ancient solutions to curvature flows
which become more and more oval for large negative times. The speed function is
a general symmetric function of the principal curvatures, homogeneous of degree
greater than one. This generalises previous work on the mean curvature flow and
other one-homogeneous curvature flows. As an auxiliary result, we prove a new
theorem on the convergence to a round point of convex rotationally symmetric
hypersurfaces satisfying a suitable constraint on the curvatures.
