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limit of probability a triangle is acute, when vertices lie on regular 2n-gon

Source: Polish MO Recond Round 1978 p5

September 9, 2024
combinatoricsgeometrycombinatorial geometry

Problem Statement

Three different points were randomly selected from the vertices of the regular 2n2n-gon. Let pn p_n be the probability of the event that the triangle with vertices at the selected points is acute-angled. Calculate lim⁡n→∞pn \lim_{n\to \infty} p_n .
Attention. We assume that all choices of three different points are equally likely.
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