# Interior Segment is small - RMO 2009 Geometry

A convex polygon $\Gamma$ is such that the distance between any two vertices of $\Gamma$ does not exceed 1.

• Prove that the distance between any two points on the boundary of $\Gamma$ does not exceed 1.
• If X and Y are two distinct points inside $\Gamma$, prove that there exists a point Z on the boundary of $\\Gamma$ such that $XZ + YZ \leq 1$

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