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Prove the inequality and when it not remains - ISL 1968

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September 23, 2010
inequalitiesgeometrypolygongeometric inequalityIMO Shortlist

Problem Statement

Let hnh_n be the apothem (distance from the center to one of the sides) of a regular nn-gon (n≥3n \geq 3) inscribed in a circle of radius rr. Prove the inequality (n+1)hn+1−nhn>r.(n + 1)h_n+1 - nh_n > r. Also prove that if rr on the right side is replaced with a greater number, the inequality will not remain true for all n≥3.n \geq 3.
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