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? | + | 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? |
== Solution == | == Solution == | ||
| − | n must satisfy these inequalities: | + | n must satisfy these [[inequality|inequalities]]: |
| + | <div style="text-align:center;"> | ||
| + | <math>4\leq n <8</math><br /> | ||
| + | <math>16\leq n<32</math><br /> | ||
| + | <math>64\leq n<128</math><br /> | ||
| + | <math>256\leq n<512</math></div> | ||
| − | <math>4 | + | 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 == | ||
| − | + | {{AIME box|year=1996|num-b=1|num-a=3}} | |
| − | + | [[Category:Intermediate Algebra Problems]] | |
Revision as of 18:54, 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. 
See also
| 1996 AIME (Problems • Answer Key • Resources) | ||
| 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 | ||
