ISL 1970 Problem 41

Let a cube of side $1$ be given. Prove that there exists a point $A$ on the surface $S$ of the cube such that every point of $S$ can be joined to $A$ by a path on $S$ of length not exceeding $2$ . Also prove that there is a point of $S$ that cannot be joined with $A$ by a path on $S$ of length less than $2$ .