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expected lenth of intersection of 5 arcs by an inscribed regular n-gon

Source: 1987 Polish MO Finals p2

January 20, 2020
combinatorial geometrycombinatoricsgeometryarcs

Problem Statement

A regular nn-gon is inscribed in a circle radius 11. Let XX be the set of all arcs PQPQ, where P,QP, Q are distinct vertices of the nn-gon. 55 elements L1,L2,...,L5L_1, L_2, ... , L_5 of XX are chosen at random (so two or more of the LiL_i can be the same). Show that the expected length of L1∩L2∩L3∩L4∩L5L_1 \cap L_2 \cap L_3 \cap L_4 \cap L_5 is independent of nn.
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