Question 166
Given 2017 positive real numbers $a_1, a_2, \dots, a_{2017}$. For each positive integer $n \gt 2017$, let
$$
a_n = \max\{a_{i_1} a_{i_2} a_{i_3} \mid i_1 + i_2 + i_3 = n, 1 \le i_1 \le i_2 \le i_3 \le n-1\}.
$$
Prove there exists some positive integers $m \le 2017$ and $N \gt 4m$ such that $a_n a_{n-4m} = a_{n-2m}^2$ for every $n \gt N$.
$$
a_n = \max\{a_{i_1} a_{i_2} a_{i_3} \mid i_1 + i_2 + i_3 = n, 1 \le i_1 \le i_2 \le i_3 \le n-1\}.
$$
Prove there exists some positive integers $m \le 2017$ and $N \gt 4m$ such that $a_n a_{n-4m} = a_{n-2m}^2$ for every $n \gt N$.