Difference between revisions of "University of South Carolina High School Math Contest/1993 Exam/Problem 17"
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− | + | <math>[\log_2 n]</math> is the largest [[integer]] <math>m</math> such that <math>2^m \leq n</math>. If we grouping the terms of our sum according to their value of <math>m</math>, the sum reduces to <math>2(1)+4(2)+8(3)+16(4)+32(5)+37(6)=2+8+24+64+160+222=480</math>. | |
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Revision as of 12:17, 1 August 2006
Problem
Let represent the greatest integer that is less than or equal to
. For example,
and
. Then what is the value of
![$[\log_2 2] + [\log_2 3] + [\log_2 4] + \cdots + [\log_2 99] + [\log_2 100] ?$](http://latex.artofproblemsolving.com/8/9/1/8911a07bc67d61a830386e643a7e75b2c80c2d57.png)
![$\mathrm{(A) \ } 480 \qquad \mathrm{(B) \ }481 \qquad \mathrm{(C) \ }482 \qquad \mathrm{(D) \ }483 \qquad \mathrm{(E) \ }484$](http://latex.artofproblemsolving.com/5/1/9/519779494da9b4d323513d74d26e6069b1936594.png)
Solution
is the largest integer
such that
. If we grouping the terms of our sum according to their value of
, the sum reduces to
.