HMMT 二月 2009 · 冲刺赛 · 第 17 题
HMMT February 2009 — Guts Round — Problem 17
题目详情
- [ 9 ] How many positive integers n ≤ 2009 have the property that b log ( n ) c is odd? 2 3 2
解析
- [ 9 ] How many positive integers n ≤ 2009 have the property that b log ( n ) c is odd? 2 Answer: 682 Solution: We wish to find n such that there is some natural number k for which 2 k − 1 ≤ log n < 2 2 k . Since n ≤ 2009 we must have k ≤ 5. This is equivalent to finding the number of positive 2 k − 1 2 k integers 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