 Solution

A simple closed curve on the sphere bounds two topological disks. Equal areas give area $2\pi$ for each, since the unit sphere has area $4\pi$. Apply Gauss-Bonnet theorem to either disk $D$, with $K=1$ and $\chi (D)=1$:A continuous geodesic curvature that were strictly positive everywhere, or strictly negative everywhere, would have a correspondingly strict sign for this integral. Therefore neither everywhere strictly positive nor everywhere strictly negative geodesic curvature is possible. Orientation reversal changes its sign but not this conclusion.