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sum a_k^4/(a_k^2+a_{k+1}^2}) >= 1/2n when sum a_i =1, a_i>0

Source: Singapore Open Math Olympiad 2001 2nd Round p2 SMO

April 2, 2020
Suminequalitiesalgebra

Problem Statement

Let nn be a positive integer, and let a1,a2,...,ana_1,a_2,...,a_n be nn positive real numbers such that a1+a2+...+an=1a_1+a_2+...+a_n = 1. Is it true that a14a12+a22+a24a22+a32+a34a32+a42+...+an−14an−12+an2+an4an2+a12≥12n\frac{a_1^4}{a_1^2+a_2^2}+\frac{a_2^4}{a_2^2+a_3^2}+\frac{a_3^4}{a_3^2+a_4^2}+...+\frac{a_{n-1}^4}{a_{n-1}^2+a_n^2}+\frac{a_n^4}{a_n^2+a_1^2}\ge \frac{1}{2n} ? Justify your answer.
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