Competition problem — Algebra — Maths Olympiad Prep

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Algebra Difficulty 6.2 National olympiad Prove it

9-196 Given that $a_{1}, a_{2}, \cdots, a_{n}$ are all positive numbers and $a_{1} \cdot a_{2} \cdots a_{n}=1$ . Prove: $\left(2+a_{1}\right)\left(2+a_{2}\right) \cdots\left(2+a_{n}\right) \geqslant 3^{n}$ .

Solution

[Proof] By the AM-GM inequality
$2+a_{k}=1+1+a_{k} \geqslant 3 a_{k}^{\frac{1}{3}}, k=1,2, \cdots, n$

Multiplying both sides of the $n$ inequalities, noting that $a_{1} \cdot a_{2} \cdots \cdot a_{n}=1$ , we obtain the desired inequality.

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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.