2021 Putnam B6

Given an ordered list of $3N$ real numbers, we can *trim* it to form a list of $N$ numbers as follows: We divide the list into $N$ groups of $3$ consecutive numbers, and within each group, discard the highest and lowest numbers, keeping only the median.

Consider generating a random number $X$ by the following procedure: Start with a list of $3^{2021}$ numbers, drawn independently and uniformly at random between 0 and 1. Then trim this list as defined above, leaving a list of $3^{2020}$ numbers. Then trim again repeatedly until just one number remains; let $X$ be this number. Let $\mu$ be the expected value of $|X - \frac{1}{2}|$. Show that

$$

\mu \geq \frac{1}{4} \left( \frac{2}{3} \right)^{2021}.

$$

给定 $3N$ 个实数的有序列表，我们可以*修剪*它以形成 $N$ 个数字的列表，如下所示：我们将列表分为 $N$ 个由 $3$ 个连续数字组成的组，并在每个组中丢弃最高和最低的数字，只保留中位数。

考虑通过以下过程生成随机数 $X$：从 $3^{2021}$ 数字的列表开始，在 0 到 1 之间独立且均匀地随机抽取。然后按照上面的定义修剪此列表，留下 $3^{2020}$ 数字的列表。然后反复修剪，直到只剩下一个数字；令 $X$ 为这个数字。令 $\mu$ 为 $|X - \frac{1}{2}|$ 的期望值。表明

$$

\mu \geq \frac{1}{4} \left( \frac{2}{3} \right)^{2021}。

$$