Putnam 1991 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

1. $p(r_i) = 0, i = 1, 2, \dots, n,$ and

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

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