Difference between revisions of "2006 AIME I Problems/Problem 13"
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== Problem == | == Problem == | ||
| − | For each even positive integer <math> x | + | For each [[even integer | even]] [[positive integer]] <math> x </math>, let <math> g(x) </math> denote the greatest power of 2 that [[divisor | 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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== See also == | == See also == | ||
| + | * [[2006 AIME I Problems/Problem 12 | Previous problem]] | ||
| + | * [[2006 AIME I Problems/Problem 14 | Next problem]] | ||
* [[2006 AIME I Problems]] | * [[2006 AIME I Problems]] | ||
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| + | [[Category:Intermediate Number Theory Problems]] | ||
Revision as of 18:34, 20 November 2006
Problem
For each even positive integer 
, let 
denote the greatest power of 2 that divides 
For example, 
and 
For each positive integer 
let 
Find the greatest integer 
less than 1000 such that 
is a perfect square.
Solution
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