PAMO 2022 Problem 3

Let $n$ be a positive integer, and $a_1, a_2, \dots, a_{2n}$ be a sequence of positive real numbers whose product is equal to $2$ . For $k = 1, 2, \dots, 2n$ , set $a_{2n + k} = a_k$ , and define $$A_k = \frac{1 + a_k + a_k a_{k + 1} + \dots + a_k a_{k + 1} \cdots a_{k + n - 2}}{1 + a_k + a_k a_{k + 1} + \dots + a_k a_{k + 1} \cdots a_{k + 2n - 2}}.$$ Suppose that $A_1, A_2, \dots, A_{2n}$ are pairwise distinct; show that exactly half of them are less than $\sqrt{2} - 1$ .