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Sum of fractions is less than 2n-1

Source: Baltic Way 2010

November 19, 2010
inequalitiesinductioninequalities proposed

Problem Statement

Let x1,x2,…,xn(n≥2)x_1, x_2, \ldots ,x_n(n\ge 2) be real numbers greater than 11. Suppose that ∣xi−xi+1∣<1|x_i-x_{i+1}|<1 for i=1,2,…,n−1i=1, 2,\ldots ,n-1. Prove that x1x2+x2x3+…+xn−1xn+xnx1<2n−1\frac{x_1}{x_2}+\frac{x_2}{x_3}+\ldots +\frac{x_{n-1}}{x_n}+\frac{x_n}{x_1}<2n-1
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