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PP_1 x PP_2 x ... x PP_n >= 2

Source: Indian Postal Coaching 2009 set 2 p3

May 26, 2020
inequalitiespolygonCyclicGeometric Inequalities

Problem Statement

Let Ω\Omega be an nn-gon inscribed in the unit circle, with vertices P1,P2,...,PnP_1, P_2, ..., P_n.
(a) Show that there exists a point PP on the unit circle such that PP1⋅PP2⋅...⋅PPn≥2PP_1 \cdot PP_2\cdot ... \cdot PP_n \ge 2.
(b) Suppose for each PP on the unit circle, the inequality PP1⋅PP2⋅...⋅PPn≤2PP_1 \cdot PP_2\cdot ... \cdot PP_n \le 2 holds. Prove that Ω\Omega is regular.
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