Difference between revisions of "1996 AIME Problems/Problem 2"
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== Solution == | == Solution == | ||
| − | n must satisfy these [[inequality|inequalities]]: | + | For integers <math>k</math>, we want <math>\lfloor \log_2 n\rfloor = 2k</math>, or <math>2k \le \log_2 < 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>): |
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<math>256\leq n<512</math></div> | <math>256\leq n<512</math></div> | ||
| − | There are 4 for the first inequality, 16 for the second, 64 for the third, and 256 for the fourth | + | 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 == | ||
Revision as of 11:53, 13 August 2008
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 | ||
