Maths Olympiad Prep

Track / Stage 6 / 218 of 400 #1218 of 1964

Problem 1218

National olympiad, first round
Algebra Difficulty 6.3 Prove it

11. Let xk>0(k=1,2,⋯ ,n),∑k=1nxˉk=1x_{k}>0(k=1,2, \cdots, n), \sum_{k=1}^{n} \bar{x}_{k}=1, prove: ∑k=1n1+xkxk⩾∏k=1nn−xk1−xk\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 and solution reproduced as published; topic, difficulty and ordering added by this site.