OMCC 2021 Problem 5

Let $n \geq 3$ be an integer and $a_1,a_2,...,a_n$ be positive real numbers such that $m$ is the smallest and $M$ is the largest of these numbers. It is known that for any distinct integers $1 \leq i,j,k \leq n$ , if $a_i \leq a_j \leq a_k$ then $a_ia_k \leq a_j^2$ . Show that

$$a_1a_2 \cdots a_n \geq m^2M^{n-2}$$

and determine when equality holds