Question 123
Let $a_1, a_2, \dots, a_{2020}$ be real numbers, not necessarily distinct. For all $n \ge 2020$, let $a_{n+1}$ be the minimal real root of the polynomial
$$
P_n(x) = x^{2n} + a_1 x^{2n-2} + a_2 x^{2n-4} + \dots + a_{n-1} x^2 + a_n,
$$
if it exists. Assume that $a_{n+1}$ exists for all $n \ge 2020$. Prove that $a_{n+1} \le a_n$ for all $n \ge 2021$.
$$
P_n(x) = x^{2n} + a_1 x^{2n-2} + a_2 x^{2n-4} + \dots + a_{n-1} x^2 + a_n,
$$
if it exists. Assume that $a_{n+1}$ exists for all $n \ge 2020$. Prove that $a_{n+1} \le a_n$ for all $n \ge 2021$.