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\frac{x_i}{\sqrt{1+x_0+x_1+\ldots+x_{i-1}}\cdot\sqrt{x_i+x_{i+1}+\ldots+x_n}}

Source: Moldova TST 1999

August 7, 2023
inequalities

Problem Statement

Let n∈N,x0=0n\in\mathbb{N}, x_0=0 and x1,x2,…,xnx_1,x_2,\ldots,x_n be postive real numbers such that x1+x2+…+xn=1x_1+x_2+\ldots+x_n=1. Show that 1≤∑i=1nxi1+x0+x1+…+xi−1⋅xi+xi+1+…+xn<π2.1\leq\sum_{i=1}^{n}\frac{x_i}{\sqrt{1+x_0+x_1+\ldots+x_{i-1}}\cdot\sqrt{x_i+x_{i+1}+\ldots+x_n}}<\frac{\pi}{2}.
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