INMO 2023 Problem 2

Suppose $a_{0}, \ldots, a_{100}$ are positive reals. Consider the following polynomial for each $k$ in $\{0,1, \ldots, 100\}$ :

$$a_{100+k} x^{100}+100 a_{99+k} x^{99}+a_{98+k} x^{98}+a_{97+k} x^{97}+\cdots+a_{2+k} x^{2}+a_{1+k} x+a_{k}$$

where indices are taken modulo 101, i.e., $a_{100+i}=a_{i-1}$ for any $i$ in $\{1,2, \ldots, 100\}$. Show that it is impossible that each of these 101 polynomials has all its roots real.