HMMT 二月 2025 · 冲刺赛 · 第 16 题

HMMT February 2025 — Guts Round — Problem 16

[9] The Cantor set is defined as the set of real numbers x such that 0 ≤ x < 1 and the digit 1 does not appear in the base-3 expansion of x . Two numbers are uniformly and independently selected at random from the Cantor set. Compute the expected value of their absolute difference. (Formally, one can pick a number x uniformly at random from the Cantor set by first picking a real number y uniformly at random from the interval [0 , 1), writing it out in binary, reading its digits as if they were in base-3, and setting x to 2 times the result.) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . HMMT February 2025, February 15, 2025 — GUTS ROUND Organization Team Team ID#

解析

[9] The Cantor set is defined as the set of real numbers x such that 0 ≤ x < 1 and the digit 1 does not appear in the base-3 expansion of x . Two numbers are uniformly and independently selected at random from the Cantor set. Compute the expected value of their absolute difference. (Formally, one can pick a number x uniformly at random from the Cantor set by first picking a real number y uniformly at random from the interval [0 , 1), writing it out in binary, reading its digits as if they were in base-3, and setting x to 2 times the result.) Proposed by: Derek Liu 2 Answer: 5 Solution: Let d be the expected value of the absolute difference. Observe that the Cantor set is made 1 up of two smaller copies of itself, each scaled down by a factor of 3. There is a chance that the two 2 selected numbers are in the same copy, in which case the expected value of their absolute difference is y 1 2+ x d . Otherwise, we can write them as and for independently and uniformly randomly selected x 3 3 3 2+( x − y ) 2 and y in the Cantor set. Their difference is , which by symmetry has expected value . Thus 3 3 1 1 1 2 2 d = · d + · = ⇒ d = . 2 3 2 3 5