1989 IMO 第 5 题

(COL 1) Consider the polynomial $p(x)=x^{n}+n x^{n-1}+a_{2} x^{n-2}+\cdots+a_{n}$ having all real roots. If $r_{1}^{16}+r_{2}^{16}+\cdots+r_{n}^{16}=n$, where the $r_{j}$ are the roots of $p(x)$, find all such roots.

(COL 1) 考虑多项式 $p(x)=x^{n}+n x^{n-1}+a_{2} x^{n-2}+\cdots+a_{n}$ 具有所有实数根。如果$r_{1}^{16}+r_{2}^{16}+\cdots+r_{n}^{16}=n$，其中$r_{j}$是$p(x)$的根，找到所有这样的根。