HMMT 二月 2025 · COMB 赛 · 第 8 题
HMMT February 2025 — COMB Round — Problem 8
题目详情
- Albert writes 2025 numbers a , . . . , a in a circle on a blackboard. Initially, each of the numbers 1 2025 is uniformly and independently sampled at random from the interval [0 , 1]. Then, each second, he simultaneously replaces a with max( a , a , a ) for all i = 1, 2, . . . , 2025 (where a = a and i i − 1 i i +1 0 2025 a = a ). Compute the expected value of the number of distinct values remaining after 100 seconds. 2026 1
解析
- Albert writes 2025 numbers a , . . . , a in a circle on a blackboard. Initially, each of the numbers 1 2025 is uniformly and independently sampled at random from the interval [0 , 1]. Then, each second, he simultaneously replaces a with max( a , a , a ) for all i = 1, 2, . . . , 2025 (where a = a and i i − 1 i i +1 0 2025 a = a ). Compute the expected value of the number of distinct values remaining after 100 seconds. 2026 1 Proposed by: Isaac Zhu 2025 Answer: 101 Solution: We can assume that the initial numbers are all distinct, since this occurs with probability