MathDB

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

3