Question 29
Suppose $a_0, \dots, a_{100}$ are positive reals. Consider the following polynomial for each $k$ in $\{0, 1, \dots, 100\}$:
$$
a_{100+k}x^{100} + 100a_{99+k}x^{99} + a_{98+k}x^{98} + a_{97+k}x^{97} + \dots + a_{2+k}x^2 + a_{1+k}x + a_k,
$$
where indices are taken modulo 101, i.e., $a_{100+i} = a_{i-1}$ for any $i$ in $\{1, 2, \dots, 100\}$. Show that it is impossible that each of these 101 polynomials has all its roots real.
$$
a_{100+k}x^{100} + 100a_{99+k}x^{99} + a_{98+k}x^{98} + a_{97+k}x^{97} + \dots + a_{2+k}x^2 + a_{1+k}x + a_k,
$$
where indices are taken modulo 101, i.e., $a_{100+i} = a_{i-1}$ for any $i$ in $\{1, 2, \dots, 100\}$. Show that it is impossible that each of these 101 polynomials has all its roots real.