Problem

Source: Chinese Mathematical Olympiad 2010 Problem 6

Tags: algebra, polynomial, number theory proposed, number theory



Suppose $a_1,a_2,a_3,b_1,b_2,b_3$ are distinct positive integers such that \[(n + 1)a_1^n + na_2^n + (n - 1)a_3^n|(n + 1)b_1^n + nb_2^n + (n - 1)b_3^n\] holds for all positive integers $n$. Prove that there exists $k\in N$ such that $ b_i = ka_i$ for $ i = 1,2,3$.