Question 134
Prove that, if positive real numbers $a_1, a_2, \dots, a_n$ have the product $1$, then
$$
\left(\frac{a_1}{a_2}\right)^{n-1} + \left(\frac{a_2}{a_3}\right)^{n-1} + \dots + \left(\frac{a_{n-1}}{a_n}\right)^{n-1} + \left(\frac{a_n}{a_1}\right)^{n-1} \geq a_1^2 + a_2^2 + \dots + a_n^2.
$$
$$
\left(\frac{a_1}{a_2}\right)^{n-1} + \left(\frac{a_2}{a_3}\right)^{n-1} + \dots + \left(\frac{a_{n-1}}{a_n}\right)^{n-1} + \left(\frac{a_n}{a_1}\right)^{n-1} \geq a_1^2 + a_2^2 + \dots + a_n^2.
$$