Question 24
Positive real numbers $a_{1}, a_{2}, \ldots, a_{n}$ with $n \geq 2$ are given such that $a_{1} a_{2} \cdots a_{n}=1$. Prove that
$$
\left(\frac{a_{1}}{a_{2}}\right)^{n-1}+\left(\frac{a_{2}}{a_{3}}\right)^{n-1}+\ldots+\left(\frac{a_{n-1}}{a_{n}}\right)^{n-1}+\left(\frac{a_{n}}{a_{1}}\right)^{n-1} \geq a_{1}^{2}+a_{2}^{2}+\ldots+a_{n}^{2}
$$
and determine when equality holds.
$$
\left(\frac{a_{1}}{a_{2}}\right)^{n-1}+\left(\frac{a_{2}}{a_{3}}\right)^{n-1}+\ldots+\left(\frac{a_{n-1}}{a_{n}}\right)^{n-1}+\left(\frac{a_{n}}{a_{1}}\right)^{n-1} \geq a_{1}^{2}+a_{2}^{2}+\ldots+a_{n}^{2}
$$
and determine when equality holds.