CMO 2020 Problem 6

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?