Subcontests
(6)\sum |P_{i-1} - P_i|^2 \le 4 for n points inside a square
There are n≥2 points in a square side 1. Show that one can label the points P1,P2,...,Pn such that ∑i=1n∣Pi−1−Pi∣2≤4, where we use cyclic subscripts, so that P0 means Pn. expected lenth of intersection of 5 arcs by an inscribed regular n-gon
A regular n-gon is inscribed in a circle radius 1. Let X be the set of all arcs PQ, where P,Q are distinct vertices of the n-gon. 5 elements L1,L2,...,L5 of X are chosen at random (so two or more of the Li can be the same). Show that the expected length of L1∩L2∩L3∩L4∩L5 is independent of n.