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point in 2n+1-gon, distance from two vertices

Source: Putnam 1989 A5

August 23, 2021
geometrycombinatoricscombinatorial geometry

Problem Statement

Show that we can find α>0\alpha>0 such that, given any point PP inside a regular 2n+12n+1-gon which is inscribed in a circle radius 11, we can find two vertices of the polygon whose distance from PP differ by less than 1n−αn3\frac1n-\frac\alpha{n^3}.
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