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Minimum value of fraction sum product

Source: China TST 2001, problem 4

May 22, 2005
algebra unsolvedalgebra

Problem Statement

For a given natural number n>3n > 3, the real numbers x1,x2,…,xn,xn+1,xn+2x_1, x_2, \ldots, x_n, x_{n + 1}, x_{n + 2} satisfy the conditions 0<x1<x2<⋯<xn<xn+1<xn+20 < x_1 < x_2 < \cdots < x_n < x_{n + 1} < x_{n + 2}. Find the minimum possible value of (∑i=1nxi+1xi)(∑j=1nxj+2xj+1)(∑k=1nxk+1xk+2xk+12+xkxk+2)(∑l=1nxl+12+xlxl+2xlxl+1)\frac{(\sum _{i=1}^n \frac{x_{i + 1}}{x_i})(\sum _{j=1}^n \frac{x_{j + 2}}{x_{j + 1}})}{(\sum _{k=1}^n \frac{x_{k + 1} x_{k + 2}}{x_{k + 1}^2 + x_k x_{k + 2}})(\sum _{l=1}^n \frac{x_{l + 1}^2 + x_l x_{l + 2}}{x_l x_{l + 1}})} and find all (n+2)(n + 2)-tuplets of real numbers (x1,x2,…,xn,xn+1,xn+2)(x_1, x_2, \ldots, x_n, x_{n + 1}, x_{n + 2}) which gives this value.
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