Difference between revisions of "1996 AIME Problems/Problem 2"
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== Problem == | == Problem == | ||
| + | For each real number <math>x</math>, let <math>\lfloor x \rfloor</math> denote the 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? | ||
| + | == Solution == | ||
| + | n must satisfy these inequalities: | ||
| + | |||
| + | |||
| + | <math>4\leq n <8</math> | ||
| + | |||
| + | <math>16\leq n<32</math> | ||
| + | |||
| + | <math>64\leq n<128</math> | ||
| + | |||
| + | <math>256\leq n<512</math> | ||
| + | |||
| + | There are 4 for the first inequality, 16 for the second, 64 for the third, and 256 for the fourth. | ||
| − | = | + | <math>4+16+64+256=340</math> |
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== See also == | == See also == | ||
* [[1996 AIME Problems/Problem 1 | Previous problem]] | * [[1996 AIME Problems/Problem 1 | Previous problem]] | ||
* [[1996 AIME Problems/Problem 3 | Next problem]] | * [[1996 AIME Problems/Problem 3 | Next problem]] | ||
* [[1996 AIME Problems]] | * [[1996 AIME Problems]] | ||
Revision as of 13:56, 24 September 2007
Problem
For each real number 
, let 
denote the greatest integer that does not exceed x. For how man positive integers 
is it true that 
and that 
is a positive even integer?
Solution
n must satisfy these inequalities:
There are 4 for the first inequality, 16 for the second, 64 for the third, and 256 for the fourth.
