Word Count : 530

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We are given: * A sequence $0 < x_1 \leq x_2 \leq \cdots \leq x_n$, with $n \geq 2$ * A condition: $$ \frac{1}{1+x_1} + \frac{1}{1+x_2} + \cdots + \frac{1}{1+x_n} = 1 $$ * We are to show: $$ \sqrt{x_1} + \sqrt{x_2} + \cdots + \sqrt{x_n} \geq (n-1)\left(\frac{1}{\sqrt{x_1}} + \cdots + \frac{1}{\sqrt{x_n}}\right) $$ --- ### Step 1: Use the given identity We are told: $$ \sum_{i=1}^n \frac{1}{1+x_i} = 1 $$ Let’s interpret this condition: Since $x_i > 0$, we have $\frac{1}{1+x_i} < 1$. Each term is positive, so this is a harmonic-type constraint. The values of $x_i$ must be such that this sum exactly adds up _________ __________ _____ ________ ____ ______ _______ _____ _________ ___ _________ ___.
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