Question 5
Given 2017 positive real numbers $a_1,a_2,\ldots,a_{2017}$, define, for every $n>2017$,
$$a_n=\max\{a_{i_1}a_{i_2}a_{i_3}:i_1+i_2+i_3=n,\ 1\le i_1\le i_2\le i_3\le n-1\}.$$
Prove that there exist positive integers $m\le2017$ and $N>4m$ such that $a_na_{n-4m}=a_{n-2m}^2$ for all $n>N$.
$$a_n=\max\{a_{i_1}a_{i_2}a_{i_3}:i_1+i_2+i_3=n,\ 1\le i_1\le i_2\le i_3\le n-1\}.$$
Prove that there exist positive integers $m\le2017$ and $N>4m$ such that $a_na_{n-4m}=a_{n-2m}^2$ for all $n>N$.