Difference between revisions of "University of South Carolina High School Math Contest/1993 Exam/Problem 17"
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== Problem == | == Problem == | ||
| + | Let <math>[x]</math> represent the greatest integer that is less than or equal to <math>x</math>. For example, <math>[2.769]=2</math> and <math>[\pi]=3</math>. Then what is the value of | ||
| − | <center><math> \mathrm{(A) \ } \qquad \mathrm{(B) \ } \qquad \mathrm{(C) \ } \qquad \mathrm{(D) \ } \qquad \mathrm{(E) \ } </math></center> | + | <center><math> [\log_2 2] + [\log_2 3] + [\log_2 4] + \cdots + [\log_2 99] + [\log_2 100] ?</math></center> |
| + | |||
| + | <center><math> \mathrm{(A) \ } 480 \qquad \mathrm{(B) \ }481 \qquad \mathrm{(C) \ }482 \qquad \mathrm{(D) \ }483 \qquad \mathrm{(E) \ }484 </math></center> | ||
== Solution == | == Solution == | ||
| + | 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>. | ||
== See also == | == See also == | ||
* [[University of South Carolina High School Math Contest/1993 Exam]] | * [[University of South Carolina High School Math Contest/1993 Exam]] | ||
Revision as of 20:15, 22 July 2006
Problem
Let ![$[x]$](http://latex.artofproblemsolving.com/b/c/e/bceb7b14e55d33a8bca29b7863ad3cdae95afce4.png)
represent the greatest integer that is less than or equal to 
. For example, ![$[2.769]=2$](http://latex.artofproblemsolving.com/e/a/0/ea00c60462a76b3a9d1bd31fafcf4a74c14c1952.png)
and ![$[\pi]=3$](http://latex.artofproblemsolving.com/e/c/a/eca8ceb8df76099894c58849611d9b55c84c9389.png)
. 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)
Solution
The sum reduces to 
.
