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Vietnam TST 2017 problem 5

Source:

March 26, 2017
algebra

Problem Statement

Given 20172017 positive real numbers a1,a2,…,a2017a_1,a_2,\dots ,a_{2017}. For each n>2017n>2017, set an=max⁡{ai1ai2ai3∣i1+i2+i3=n,1≤i1≤i2≤i3≤n−1}.a_n=\max\{ a_{i_1}a_{i_2}a_{i_3}|i_1+i_2+i_3=n, 1\leq i_1\leq i_2\leq i_3\leq n-1\}. Prove that there exists a positive integer m≤2017m\leq 2017 and a positive integer N>4mN>4m such that anan−4m=an−2m2a_na_{n-4m}=a_{n-2m}^2 for every n>Nn>N.
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