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Inequality on a plane

Source: Tuymaada 2023 Senior P6

July 8, 2023
inequalitiesgeometryPlaneTuymaadaGeometry inequality

Problem Statement

In the plane nn segments with lengths a1,a2,…,ana_1, a_2, \dots , a_n are drawn. Every ray beginning at the point OO meets at least one of the segments. Let hih_i be the distance from OO to the ii-th segment (not the line!) Prove the inequality a1h1+a2h2+…+aihi⩾2π.\frac{a_1}{h_1}+\frac{a_2}{h_2} + \ldots + \frac{a_i}{h_i} \geqslant 2 \pi.
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