2007 Putnam B5

Let $k$ be a positive integer. Prove that there exist polynomials

$P_0(n), P_1(n), \dots, P_{k-1}(n)$ (which may depend on $k$) such that

for any integer $n$,

$$

\left\lfloor \frac{n}{k} \right\rfloor^k = P_0(n) + P_1(n) \left\lfloor

\frac{n}{k} \right\rfloor + \cdots + P_{k-1}(n) \left\lfloor \frac{n}{k}

\right\rfloor^{k-1}.

$$

($\lfloor a \rfloor$ means the largest integer $\leq a$.)

令 $k$ 为正整数。证明多项式存在

$P_0(n), P_1(n), \dots, P_{k-1}(n)$ (可能取决于 $k$)，使得

对于任意整数$n$，

$$

\left\lfloor \frac{n}{k} \right\rfloor^k = P_0(n) + P_1(n) \left\lfloor

\frac{n}{k} \right\rfloor + \cdots + P_{k-1}(n) \left\lfloor \frac{n}{k}

\right\rfloor^{k-1}。

$$

($\lfloor a \rfloor$ 表示最大整数 $\leq a$。)