1999 Putnam A2

Let $p(x)$ be a polynomial that is nonnegative for all real $x$. Prove that

for some $k$, there are polynomials $f_1(x),\dots,f_k(x$) such that

$$p(x) = \sum_{j=1}^k (f_j(x))^2.$$

令 $p(x)$ 为一个对于所有实数 $x$ 均为非负的多项式。证明

对于某些 $k$，有多项式 $f_1(x),\dots,f_k(x$) 使得

$$p(x) = \sum_{j=1}^k (f_j(x))^2。$$