2011 Putnam B6

Let $p$ be an odd prime. Show that for at least $(p+1)/2$ values of $n$ in $\{0,1,2,\dots,p-1\}$,

$$

\sum_{k=0}^{p-1} k! n^k \qquad \text{is not divisible by $p$.}

$$

令 $p$ 为奇素数。证明对于 $\{0,1,2,\dots,p-1\}$ 中 $n$ 的至少 $(p+1)/2$ 值，

$$

\sum_{k=0}^{p-1} k！ n^k \qquad \text{不能被 $p$ 整除。}

$$