Let $a_0, a_1, a_2,\ldots $ be a sequence of real numbers satisfying $a_0=1$ and $a_n=a_{\lfloor 7n/9\rfloor}+a_{\lfloor n/9\rfloor}$ for $n=1, 2,\ldots $ Prove that there exists a positive integer $k$ with $a_k<\frac{k}{2001!}$.
Source: Baltic Way 2001
Tags: floor function, algebra proposed, algebra
Let $a_0, a_1, a_2,\ldots $ be a sequence of real numbers satisfying $a_0=1$ and $a_n=a_{\lfloor 7n/9\rfloor}+a_{\lfloor n/9\rfloor}$ for $n=1, 2,\ldots $ Prove that there exists a positive integer $k$ with $a_k<\frac{k}{2001!}$.