HMMT 二月 2009 · 冲刺赛 · 第 17 题

英文解析

17. [ 9 ] How many positive integers n ≤ 2009 have the property that b log ( n ) c is odd?
    Answer: 6822
    Solution: We wish to find n such that there is some natural number k for which 2 k − 1 ≤ log n <
    2 k . Since n ≤ 2009 we must have k ≤ 5. This is equivalent to finding the number of positive 2
    2 k − 1 2 kintegers n ≤ 2009 satisfying 2 ≤ n < 2 for some k ≤ 5, so the number of such integers is
    3 5 7 9
    2 + 2 + 2 + 2 + 2 = 682.
    3 2