Question 77
Let $a_1, a_2, \dots$ be an infinite sequence of distinct non-zero real numbers for which $\frac{a_{i+1}}{a_i} + \frac{a_{i+1}}{a_i}$ takes the same value lying in between 0 and 2 for each $i \ge 1$. Express in terms of $a_1, a_2, a_3$ the smallest number $c$ satisfying the following condition:
Condition: For any pair of positive integers $x, y$ with $x \lt y$,
$$
\frac{a_x a_{x+1} + a_{x+1} a_{x+2} + \dots + a_{y-1} a_y}{a_x a_y} \le c \quad \text{holds.}
$$
Condition: For any pair of positive integers $x, y$ with $x \lt y$,
$$
\frac{a_x a_{x+1} + a_{x+1} a_{x+2} + \dots + a_{y-1} a_y}{a_x a_y} \le c \quad \text{holds.}
$$