Competition problem — Algebra — Maths Olympiad Prep

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Algebra Difficulty 6.4 National olympiad Prove it

47. Given that $x_{1}, x_{2}, \cdots, x_{n}$ are positive numbers, and $x_{1} x_{2} \cdots x_{n}=1$ , prove: $\frac{1}{x_{1}\left(x_{1}+1\right)}$ $\frac{1}{x_{2}\left(x_{2}+1\right)}+\cdots+\frac{1}{x_{n}\left(x_{n}+1\right)} \geqslant \frac{n}{2} .(2010$ Moldova National Training Team Problem)

Solution

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