Question 133
Let $a_1, a_2, a_3, b_1, b_2, b_3$ be pairwise distinct positive integers such that
$$
(n+1)a_1^2 + n a_2^2 + (n-1)a_3^2 \mid (n+1)b_1^2 + n b_2^2 + (n-1)b_3^2
$$
holds for all positive integers $n$. Prove that there exists a positive integer $k$ such that $b_i = k a_i$ for all $i = 1, 2, 3$.
$$
(n+1)a_1^2 + n a_2^2 + (n-1)a_3^2 \mid (n+1)b_1^2 + n b_2^2 + (n-1)b_3^2
$$
holds for all positive integers $n$. Prove that there exists a positive integer $k$ such that $b_i = k a_i$ for all $i = 1, 2, 3$.