New Zealand 2013 TST problem:
Let $r$ and $s$ be positive integers. Define $a_0 = 0$, $a_1 = 1$, and $a_n = ra_{n-1} + sa_{n-2}$ for $n \geq 2$. Let $f_n = a_1a_2\cdots a_n$.
Prove that $\dfrac{f_n}{f_kf_{n-k}}$ is an integer for all integers $n$ and $k$ such that $0 < k < n$.
In other words,we want to show that $$\prod_{i=1}^{k}a_{i}|\prod_{i=1}^{k}a_{t+i}$$ for all $t\ge 0$.
maybe this post