TITLE: (Tournament of towns 1994) Prove the inequality
QUESTION [7 upvotes]: Let $a_1,a_2,\ldots,a_n$ be real positive numbers. Prove that $$\left(1+\frac{a_1^2}{a_2}\right)\left(1+\frac{a_2^2}{a_3}\right) \cdots \left(1+\frac{a_n^2}{a_1}\right) \geq(1+a_1)(1+a_2) \cdots (1+a_n)$$

REPLY [12 votes]: By Cauchy-Schwarz inequality, we have the following:
$$(1+a_2)\left(1+\frac{a_1^2}{a_2}\right)\geq (1+a_1)^2$$
$$(1+a_3)\left(1+\frac{a_2^2}{a_3}\right)\geq (1+a_2)^2$$
$$\vdots$$
$$(1+a_1)\left(1+\frac{a_n^2}{a_1}\right)\geq (1+a_n)^2$$
from where we have: $$\left(1+\frac{a_1^2}{a_2}\right)\left(1+\frac{a_2^2}{a_3}\right)\cdots\left(1+\frac{a_n^2}{a_1}\right)\prod_{i=1}^n (1+a_i)\geq \prod_{i=1}^n (1+a_i)^2.$$ By division with $\prod_{i=1}^n (1+a_i)$ we get 
$$\left(1+\frac{a_1^2}{a_2}\right)\left(1+\frac{a_2^2}{a_3}\right)\cdots\left(1+\frac{a_n^2}{a_1}\right)\geq \prod_{i=1}^n (1+a_i).$$