2019 Putnam A3

Given real numbers $b_0, b_1, \dots, b_{2019}$ with $b_{2019} \neq 0$, let $z_1,z_2,\dots,z_{2019}$ be

the roots in the complex plane of the polynomial

$$

P(z) = \sum_{k=0}^{2019} b_k z^k.

$$

Let $\mu = (|z_1| + \cdots + |z_{2019}|)/2019$ be the average of the distances from $z_1,z_2,\dots,z_{2019}$ to the origin. Determine the largest constant $M$ such that $\mu \geq M$ for all choices of $b_0,b_1,\dots, b_{2019}$ that satisfy

$$

1 \leq b_0 < b_1 < b_2 < \cdots < b_{2019} \leq 2019.

$$

给定实数 $b_0, b_1, \dots, b_{2019}$ 且 $b_{2019} \neq 0$，设 $z_1,z_2,\dots,z_{2019}$ 为

多项式复平面中的根

$$

P(z) = \sum_{k=0}^{2019} b_k z^k。

$$

设 $\mu = (|z_1| + \cdots + |z_{2019}|)/2019$ 为从 $z_1,z_2,\dots,z_{2019}$ 到原点的距离的平均值。确定最大常数$M$，使得$b_0,b_1,\dots, b_{2019}$的所有选择满足$\mu \geq M$

$$

1 \leq b_0 < b_1 < b_2 < \cdots < b_{2019} \leq 2019。

$$