HMMT 二月 2002 · 冲刺赛 · 第 37 题

HMMT February 2002 — Guts Round — Problem 37

[8] Call a positive integer “mild” if its base-3 representation never contains the digit 2. 2 How many values of n (1 ≤ n ≤ 1000) have the property that n and n are both mild?

解析

Call a positive integer “mild” if its base-3 representation never contains the digit 2. 2 How many values of n (1 ≤ n ≤ 1000) have the property that n and n are both mild? Solution: 7 Such a number, which must consist entirely of 0’s and 1’s in base 3, can a b never have more than one 1. Indeed, if n = 3 + 3 + higher powers where b > a , then 2 2 a a + b n = 3 + 2 · 3 + higher powers which will not be mild. On the other hand, if n does just 2 have one 1 in base 3, then clearly n and n are mild. So the values of n ≤ 1000 that work 0 1 6 are 3 , 3 , . . . , 3 ; there are 7 of them. 9