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Dividing the perimeter with points

Source: 239 2010 S7

July 29, 2020

Problem Statement

You are given a convex polygon with perimeter 243+4π24\sqrt{3} + 4\pi. If there exists a pair of points dividing the perimeter in half such that the distance between them is equal to 2424, Prove that there exists a pair of points dividing the perimeter in half such that the distance between them does not exceed 1212.
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