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 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 == | ||
| + | For integers <math>k</math>, we want <math>\lfloor \log_2 n\rfloor = 2k</math>, or <math>2k \le \log_2 n < 2k+1 \Longrightarrow 2^{2k} \le n < 2^{2k+1}</math>. Thus, <math>n</math> must satisfy these [[inequality|inequalities]] (since <math>n < 1000</math>): | ||
| + | |||
| + | <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> | ||
| + | |||
| + | There are <math>4</math> for the first inequality, <math>16</math> for the second, <math>64</math> for the third, and <math>256</math> for the fourth, so the answer is <math>4+16+64+256=\boxed{340}</math>. | ||
== See also == | == See also == | ||
| − | + | {{AIME box|year=1996|num-b=1|num-a=3}} | |
| + | |||
| + | [[Category:Intermediate Algebra Problems]] | ||
| + | {{MAA Notice}} | ||
Latest revision as of 18:31, 4 July 2013
Problem
For each real number 
, let 
denote the greatest integer that does not exceed x. For how many positive integers 
is it true that 
and that 
is a positive even integer?
Solution
For integers 
, we want 
, or 
. Thus, must satisfy these inequalities (since 
):
There are 
for the first inequality, 
for the second, 
for the third, and 
for the fourth, so the answer is 
.
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 | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
