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min (q_r(C)+q_a(C)) \le \frac{1}{32} {n \choose 4}, coloring inequality

Source: Romania IMO TST 1991 p3

February 19, 2020

inequalitiesColoringcombinatoricspolygon

Problem Statement

Let

C

be a coloring of all edges and diagonals of a convex

n

−gon in red and blue (in Romanian, rosu and albastru). Denote by

q_r(C)

(resp.

q_a(C)

) the number of quadrilaterals having all its edges and diagonals red (resp. blue). Prove:

\underset{C}{min} (q_r(C)+q_a(C)) \le \frac{1}{32} {n \choose 4}