2020 CMO 第 6 题

Does there exist positive reals $a_0, a_1,\ldots ,a_{19}$ , such that the polynomial $P(x)=x^{20}+a_{19}x^{19}+\ldots +a_1x+a_0$ does not have any real roots, yet all polynomials formed from swapping any two coefficients $a_i,a_j$ has at least one real root?

是否存在正实数 $a_0, a_1,\ldots ,a_{19}$ ，使得多项式 $P(x)=x^{20}+a_{19}x^{19}+\ldots +a_1x+a_0$ 没有任何实根，但通过交换任意两个系数 $a_i,a_j$ 形成的所有多项式都至少有一个实根？