Competition problem — Algebra — Maths Olympiad Prep

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Algebra Difficulty 6.1 National olympiad Prove it

5 Let $a_{i}, b_{i} \in \mathbf{R}^{+}(i=1,2, \cdots, n)$ , prove:
$\left[\prod_{i=1}^{n}\left(a_{i}+b_{i}\right)\right]^{\frac{1}{n}} \geqslant\left(\prod_{i=1}^{n} a_{i}\right)^{\frac{1}{n}}+\left(\prod_{i=1}^{n} b_{i}\right)^{\frac{1}{n}}$

where $\prod_{i=1}^{n} a_{i}=a_{1} a_{2} \cdots a_{n}$ .

Solution

5. Spitting $=\left(\Pi \frac{a_{i}}{a_{i}+b_{i}}\right)^{\frac{1}{1}}+\left(\Pi \frac{b_{i}}{a_{i}+b_{i}}\right)^{\frac{1}{n}} \leqslant \frac{1}{n}\left(\sum \frac{a_{i}}{a_{i}+b_{i}}+\sum \frac{b_{i}}{a_{i}+b_{i}}\right)=$
1. So left $\geqslant$ .

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.