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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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