Difference between revisions of "2006 AIME I Problems/Problem 13"

 
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== Problem ==
 
== Problem ==
 
For each even positive integer <math> x, </math> let <math> g(x) </math> denote the greatest power of 2 that divides <math> x. </math> For example, <math> g(20)=4 </math> and <math> g(16)=16. </math> For each positive integer <math> n, </math> let <math> S_n=\sum_{k=1}^{2^{n-1}}g(2k). </math> Find the greatest integer <math> n </math> less than 1000 such that <math> S_n </math> is a perfect square.
 
For each even positive integer <math> x, </math> let <math> g(x) </math> denote the greatest power of 2 that divides <math> x. </math> For example, <math> g(20)=4 </math> and <math> g(16)=16. </math> For each positive integer <math> n, </math> let <math> S_n=\sum_{k=1}^{2^{n-1}}g(2k). </math> Find the greatest integer <math> n </math> less than 1000 such that <math> S_n </math> is a perfect square.
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== Solution ==
 
== Solution ==
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== See also ==
 
== See also ==
* [[2006 AIME I]]
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* [[2006 AIME I Problems]]

Revision as of 12:15, 30 June 2006

Problem

For each even positive integer $x,$ let $g(x)$ denote the greatest power of 2 that divides $x.$ For example, $g(20)=4$ and $g(16)=16.$ For each positive integer $n,$ let $S_n=\sum_{k=1}^{2^{n-1}}g(2k).$ Find the greatest integer $n$ less than 1000 such that $S_n$ is a perfect square.



Solution

See also