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sum x_k(2x_k - x_{k+1} - x_{k+2}) / ( x_{k+1} + x_{k+2} ) >= 0

Source: 2009 Cuba 2.7

August 27, 2024
algebrainequalities

Problem Statement

Let x1,x2,...,xnx_1, x_2, ..., x_n be positive reals. Prove that ∑k=1nxk(2xk−xk+1−xk+2)xk+1+xk+2≥0\sum_{k=1}^n \frac{x_k(2x_k - x_{k+1} - x_{k+2})}{x_{k+1} + x_{k+2}} \ge 0 In the sum, cyclic indices have been taken, that is, xn+1=x1x_{n+1} = x_1 and xn+2=x2x_{n+2} = x_2.
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