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Difference between revisions of "University of South Carolina High School Math Contest/1993 Exam/Problem 17"

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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>
 
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>
  
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<center><math> \mathrm{(A) \ } 480 \qquad \mathrm{(B) \ }481 \qquad \mathrm{(C) \ }482 \qquad \mathrm{(D) \ }483 \qquad \mathrm{(E) \ }484 </math></center>
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<math> \mathrm{(A) \ } 480 \qquad \mathrm{(B) \ }481 \qquad \mathrm{(C) \ }482 \qquad \mathrm{(D) \ }483 \qquad \mathrm{(E) \ }484 </math>
  
  

Latest revision as of 17:14, 1 August 2006

Problem

Let $[x]$ represent the greatest integer that is less than or equal to $x$. For example, $[2.769]=2$ and $[\pi]=3$. Then what is the value of $[\log_2 2] + [\log_2 3] + [\log_2 4] + \cdots + [\log_2 99] + [\log_2 100] ?$

$\mathrm{(A) \ } 480 \qquad \mathrm{(B) \ }481 \qquad \mathrm{(C) \ }482 \qquad \mathrm{(D) \ }483 \qquad \mathrm{(E) \ }484$


Solution

$[\log_2 n]$ is the largest integer $m$ such that $2^m \leq n$. If we grouping the terms of our sum according to their value of $m$, the sum reduces to $2(1)+4(2)+8(3)+16(4)+32(5)+37(6)=2+8+24+64+160+222=480 \Rightarrow A$.

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