Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

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 $2(1)+4(2)+8(3)+16(4)+32(5)+37(6)=2+8+24+64+160+222=480 \Rightarrow A$.

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=University_of_South_Carolina_High_School_Math_Contest/1993_Exam/Problem_17&oldid=8949"