University of South Carolina High School Math Contest/1993 Exam/Problem 17

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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