IR-MO 2004 Problem 2

$A$ is a compact convex set in plane. Prove that there exists a point $O \in A$ , such that for every line $XX'$ passing through $O$ , where $X$ and $X'$ are boundary points of $A$ , then $$\frac12 \leq \frac {OX}{OX'} \leq 2.$$