HMMT November 2024 U35

There are 1024 players, ranked from 1 (most skilled) to 1024 (least skilled), participating in a single elimination tournament. In each of the 10 rounds, the remaining players are paired uniformly at random. In each match, the player with a lower rank always wins, and the loser is eliminated from the tournament.

For each positive integer $n \in[1,1024]$, let $f(n)$ be the expected number of rounds that the participant with rank $n$ participates in. Estimate the minimum positive integer $N$ such that $f(N)<2$.

Submit a positive integer $E$. If the correct answer is $A$, you will receive max $\left(0,20-\left\lfloor\frac{|E-A|}{2}\right\rfloor\right)$ points. Proposed by: Aaron Guo