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TOT 020 1982 Spring S1 sum {x_k}{x_{k-1}+x_{k+1}} >=2, sharp inequality

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August 17, 2019
inequalitiesalgebra

Problem Statement

(a) Prove that for any positive numbers x1,x2,...,xkx_1,x_2,...,x_k (k>3k > 3), x1xk+x2+x2x1+x3+...+xkxk−1+x1≥2\frac{x_1}{x_k+x_2}+ \frac{x_2}{x_1+x_3}+...+\frac{x_k}{x_{k-1}+x_1}\ge 2 (b) Prove that for every kk this inequality cannot be sharpened, i.e. prove that for every given kk it is not possible to change the number 22 in the right hand side to a greater number in such a way that the inequality remains true for every choice of positive numbers x1,x2,...,xkx_1,x_2,...,x_k.
(A Prokopiev)
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