RMO 2009 Problem 5

A convex polygon is such that the distance between any two vertices does not exceed $1$ . $(i)$ Prove that the distance between any two points on the boundary of the polygon does not exceed $1$ . $(ii)$ If $X$ and $Y$ are two distinct points inside the polygon, prove that there exists a point $Z$ on the boundary of the polygon such that $XZ + YZ\le1$ .