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Cyclic inequality a_1 (b_1 + a_2) + a_2 (b_2 + a_3) + ... <1

Source: 7th German TST 2005, problem 2; Japan Mathematical Olympiad Finals 2002 , Problem 4

May 30, 2005
inequalitiesn-variable inequalityGermanyTST

Problem Statement

Let n n be a positive integer such that n≥3 n\geq 3. Let a1 a_1, a2 a_2, ..., an a_n and b1 b_1, b2 b_2, ..., bn b_n be 2n 2n positive real numbers satisfying the equations a_1 \plus{} a_2 \plus{} ... \plus{} a_n \equal{} 1,   \text{and}   b_1^2 \plus{} b_2^2 \plus{} ... \plus{} b_n^2 \equal{} 1. Prove the inequality a_1\left(b_1 \plus{} a_2\right) \plus{} a_2\left(b_2 \plus{} a_3\right) \plus{} ... \plus{} a_{n \minus{} 1}\left(b_{n \minus{} 1} \plus{} a_n\right) \plus{} a_n\left(b_n \plus{} a_1\right) < 1.
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