Putnam 1994 B5

For any real number $\alpha$, define the function $f_{\alpha}(x) = \lfloor \alpha x \rfloor$. Let $n$ be a positive integer. Show that there exists an $\alpha$ such that for $1 \leq k \leq n$, $$f_\alpha^k(n^2) = n^2 - k = f_{\alpha^k}(n^2).$$