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