IMO Shortlist 1968 problem 5


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Let h_n be the apothem (distance from the center to one of the sides) of a regular n-gon (n \geq 3) inscribed in a circle of radius r. Prove the inequality
(n + 1)h_n+1 - nh_n > r.
Also prove that if r on the right side is replaced with a greater number, the inequality will not remain true for all n \geq 3.
Izvor: Međunarodna matematička olimpijada, shortlist 1968