1994 Putnam 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).

$$

对于任意实数$\alpha$，定义函数$f_{\alpha}(x)

= \lfloor \alpha x \rfloor$。令$n$ 为正整数。表明

存在 $\alpha$ 使得对于 $1 \leq k \leq n$，

$$

f_\alpha^k(n^2) = n^2 - k = f_{\alpha^k}(n^2)。

$$