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1/h_1+1h_2+...+1/h_n> 2\pi /a inside a regular n-gon

Source: Estonia IMO TST 2001 p2

March 26, 2020

regular polygonGeometric Inequalitiesgeometrydistance

Problem Statement

Point

X

is taken inside a regular

n

-gon of side length

a

. Let

h_1,h_2,...,h_n

be the distances from

X

to the lines defined by the sides of the

n

-gon. Prove that

\frac{1}{h_1}+\frac{1}{h_2}+...+\frac{1}{h_n}>\frac{2\pi}{a}