HMMT 二月 2025 · COMB 赛 · 第 1 题

HMMT February 2025 — COMB Round — Problem 1

Compute the number of ways to arrange the numbers 1, 2, 3, 4, 5, 6, and 7 around a circle such that the product of every pair of adjacent numbers on the circle is at most 20. (Rotations and reflections count as different arrangements.)

解析

For clarity, we denote the value of a after t seconds as a . The index i is taken mod 2025. i i,t 2025 In general, after k < 1012 seconds, we claim the expected number of distinct values remaining is . k +1 To show this, we first prove that for any remaining value, its appearances are consecutive. Indeed, note that for all i and k , a = max( a , a , a ) = max( a , . . . , a ) = · · · = max( a , . . . , a ) . i,k i − 1 ,k − 1 i,k − 1 i +1 ,k − 1 i − 2 ,k − 2 i +2 ,k − 2 i − k, 0 i + k, 0 Given an initial number a , let j and j be the smallest positive integers such that a > c, 0 1 2 c − j , 0 1 a and a > a . As the initial numbers are distinct, we conclude a = a if and only if c, 0 c + j , 0 c, 0 i,k c, 0 2 { i − k, i − k + 1 , . . . , i + k } contains c but neither c − j nor c + j (mod 2025). The indices i that 1 2 satisfy this are clearly consecutive. Now consider the indicator variables ( 1 if a ̸ = a i,k i +1 ,k 1 = i 0 otherwise for 1 ≤ i ≤ 2025. Note that the number of distinct values on the board after k seconds is simply P 2025