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Cyclic inequality

Source: Japan Mathematical Olympiad Finals 2002 , Problem 4

March 22, 2006
inequalitiesinequalities proposed

Problem Statement

Let n≥3n\geq 3 be natural numbers, and let a1, a2, ⋯ , an,  b1, b2, ⋯ , bna_1,\ a_2,\ \cdots,\ a_n,\ \ b_1,\ b_2,\ \cdots,\ b_n be positive numbers such that a1+a2+⋯+an=1, b12+b22+⋯+bn2=1.a_1+a_2+\cdots +a_n=1,\ b_1^2+b_2^2+\cdots +b_n^2=1. Prove that a1(b1+a2)+a2(b2+a3)+⋯+an(bn+a1)<1.a_1(b_1+a_2)+a_2(b_2+a_3)+\cdots +a_n(b_n+a_1)<1.
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