AIME 1996 · 第 2 题

AIME 1996 — Problem 2

专题: Contest Math
难度: L4
来源: AIME

题目详情

Problem

For each real number $x$ , let $\lfloor x \rfloor$ denote the greatest integer that does not exceed x. For how many positive integers $n$ is it true that $n<1000$ and that $\lfloor \log_{2} n \rfloor$ is a positive even integer?

解析

Solution

For integers $k$ , we want $\lfloor \log_2 n\rfloor = 2k$ , or $2k \le \log_2 n < 2k+1 \Longrightarrow 2^{2k} \le n < 2^{2k+1}$ . Thus, $n$ must satisfy these inequalities (since $n < 1000$ ):

$4\leq n <8$ $16\leq n<32$ $64\leq n<128$

256\leq n<512

There are $4$ for the first inequality, $16$ for the second, $64$ for the third, and $256$ for the fourth, so the answer is $4+16+64+256=\boxed{340}$ .