1991 Putnam A3

Find all real polynomials $p(x)$ of degree $n \geq 2$ for which there
exist real numbers $r_1 < r_2 < \cdots < r_n$ such that

\item $p(r_i) = 0, \qquad i = 1, 2, \dots, n,$ and
\item $p' \left( \frac{r_i + r_{i+1}}{2} \right) = 0 \qquad i = 1, 2,
\dots, n-1,$

where $p'(x)$ denotes the derivative of $p(x)$.

查找 $n \geq 2$ 次的所有实多项式 $p(x)$，其中

存在实数 $r_1 < r_2 < \cdots < r_n$ 使得

\item $p(r_i) = 0, \qquad i = 1, 2, \dots, n,$ 和

\item $p' \left( \frac{r_i + r_{i+1}}{2} \right) = 0 \qquad i = 1, 2,

\点，n-1，$

其中$p'(x)$表示$p(x)$的导数。