1996 AIME Problems/Problem 2

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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?

Solution

n must satisfy these inequalities:

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

$256\leq n<512$

There are 4 for the first inequality, 16 for the second, 64 for the third, and 256 for the fourth. $4+16+64+256=340$

See also

1996 AIME (ProblemsAnswer KeyResources)
Preceded by
Problem 1
Followed by
Problem 3
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All AIME Problems and Solutions
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