Track / Stage 6 / 274 of 400 #1274 of 1964

Problem 1274

National Olympiad, first round

Algebra Difficulty 6.3 Prove it

Let $x_{k}>0(k=1,2, \cdots, n), \sum_{k=1}^{n} \bar{x}_{k}=1$ , prove: $\sum_{k=1}^{n} \frac{1+x_{k}}{x_{k}} \geqslant \prod_{k=1}^{n} \frac{n-x_{k}}{1-x_{k}}$ . (2006 China National Training Team Problem)

This one wants a proof. Work it on paper, then read the official solution and mark yourself. Be honest about it: the record is only any use to you if it is.

Official solution

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement reproduced verbatim; metadata (topic, difficulty, ordering) added by this project.