HMMT 二月 2026 · COMB 赛 · 第 4 题

HMMT February 2026 — COMB Round — Problem 4

Sarunyu has a stick of length 1 with one endpoint marked in red. Every minute, he picks one of his sticks uniformly at random and breaks it into two halves of equal length. Compute the expected length of the stick with the red endpoint after 5 minutes. 9

解析

Sarunyu has a stick of length 1 with one endpoint marked in red. Every minute, he picks one of his sticks uniformly at random and breaks it into two halves of equal length. Compute the expected length of the stick with the red endpoint after 5 minutes. Proposed by: Derek Liu 10 ( ) 63 5 Answer: = 10 2 256 Solution: Let E denote the expected length of the stick with the red endpoint after n minutes. n Every minute, Sarunyu breaks one stick into two, so right before the n th minute, Sarunyu has n sticks. 1 The probability that he selects the stick with the red endpoint is . Therefore, if the stick with the n x red endpoint had length x before the n th minute, then it will have length after the n th minute with 2 1 th n − 1 probability , and length x after the n minute with probability . Hence, the expected length of n n th 1 1 n − 1 2 n − 1 the stick after the n minute is ( · + ) x = x , regardless of the value of x . By linearity of n 2 n 2 n expectation, this means 2 n − 1 E = E . n n − 1 2 n We conclude that the expected length after 5 minutes is ( ) ( ) ( ) 1 3 9 63 E = · · · = . 5 2 4 10 256 9