5. (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.
5. According to the Cauchy-Schwarz inequality,
$$
\left(\sum_{i=1}^{n} a_{i}\right)^{2} \leq\left(\sum_{i=1}^{n} a_{i}^{2}\right)\left(\sum_{i=1}^{n} 1^{2}\right)=n\left(\sum_{i=1}^{n} a_{i}^{2}\right) .
$$

Since $r_{1}+\cdots+r_{n}=-n$, applying this inequality we obtain $r_{1}^{2}+\ldots+r_{n}^{2} \geq n$. and applying it three more times, we obtain
$$
r_{1}^{16}+\cdots+r_{n}^{16} \geq n,
$$
with equality if and only if $r_{1}=r_{2}=\ldots=r_{n}=-1$ and $p(x)=(x+1)^{n}$.