Question 69
Find all sequences $(a_n)_{n \in \mathbb{N}}$ of positive integers such that for every $n \ge 3$ we have:
$$
\frac{1}{a_1 a_3} + \frac{1}{a_2 a_4} + \frac{1}{a_3 a_5} + \dots + \frac{1}{a_{n-2} a_n} = 1 - \frac{1}{a_1^2 + a_2^2 + \dots + a_{n-1}^2}
$$
$$
\frac{1}{a_1 a_3} + \frac{1}{a_2 a_4} + \frac{1}{a_3 a_5} + \dots + \frac{1}{a_{n-2} a_n} = 1 - \frac{1}{a_1^2 + a_2^2 + \dots + a_{n-1}^2}
$$