Competition problem — Algebra — Maths Olympiad Prep

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Algebra Difficulty 6.4 National olympiad Prove it

Example 3 Let $x_{i}>0, i=1,2, \cdots, n(n \geqslant 3)$ , and
$\begin{aligned} & \sum_{i=1}^{n} x_{t}=1 \\ \text { then } & \prod_{i=1}^{n}\left(\frac{1}{x_{i}}-x_{i}\right) \geqslant\left(n-\frac{1}{n}\right)^{n} . \end{aligned}$

Solution

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