Difference between revisions of "1996 AIME Problems/Problem 2"

m (cat, fmt)
m (Problem)
Line 1: Line 1:
 
== Problem ==
 
== Problem ==
For each real number <math>x</math>, let <math>\lfloor x \rfloor</math> denote the [[greatest integer function|greatest integer]] that does not exceed x. For how man positive integers <math>n</math> is it true that <math>n<1000</math> and that <math>\lfloor \log_{2} n \rfloor</math> is a positive even integer?
+
For each real number <math>x</math>, let <math>\lfloor x \rfloor</math> denote the [[greatest integer function|greatest integer]] that does not exceed x. For how many positive integers <math>n</math> is it true that <math>n<1000</math> and that <math>\lfloor \log_{2} n \rfloor</math> is a positive even integer?
  
 
== Solution ==
 
== Solution ==

Revision as of 13:52, 12 November 2007

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
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
All AIME Problems and Solutions
Invalid username
Login to AoPS