Difference between revisions of "1996 AIME Problems/Problem 2"
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| − | 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>): | + | 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>): |
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[[Category:Intermediate Algebra Problems]] | [[Category:Intermediate Algebra Problems]] | ||
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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. 
