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 | + | 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> |
− | + | <math> \mathrm{(A) \ } 480 \qquad \mathrm{(B) \ }481 \qquad \mathrm{(C) \ }482 \qquad \mathrm{(D) \ }483 \qquad \mathrm{(E) \ }484 </math> | |
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== Solution == | == Solution == | ||
− | + | <math>[\log_2 n]</math> is the largest [[integer]] <math>m</math> such that <math>2^m \leq n</math>. If we grouping the terms of our sum according to their value of <math>m</math>, the sum reduces to <math>2(1)+4(2)+8(3)+16(4)+32(5)+37(6)=2+8+24+64+160+222=480 \Rightarrow A</math>. | |
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Latest revision as of 17:14, 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
Solution
is the largest integer such that . If we grouping the terms of our sum according to their value of , the sum reduces to .