Difference between revisions of "University of South Carolina High School Math Contest/1993 Exam/Problem 17"
m |
|||
Line 1: | Line 1: | ||
== 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 | + | 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 <math> [\log_2 2] + [\log_2 3] + [\log_2 4] + \cdots + [\log_2 99] + [\log_2 100] ? </math> |
− | <center><math> | + | <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 == |
Revision as of 17:13, 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
![$\mathrm{(A) \ } 480 \qquad \mathrm{(B) \ }481 \qquad \mathrm{(C) \ }482 \qquad \mathrm{(D) \ }483 \qquad \mathrm{(E) \ }484$](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
.