HMMT 二月 2002 · 冲刺赛 · 第 37 题

英文解析

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