IMC 1996 · Problem 4

Let $B$ be a bounded closed convex symmetric (with respect to the origin) set in $\mathbb{R}^2$ with boundary the curve $\Gamma$. Let $B$ have the property that the ellipse of maximal area contained in $B$ is the disc $D$ of radius $1$ centered at the origin with boundary the circle $C$. Prove that $A \cap \Gamma \ne \emptyset$ for any arc $A$ of $C$ of length $l(A) > \dfrac{\pi}{2}$.