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

Revision as of 14:48, 25 September 2007

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

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 == Problem ==
-How many [[integer]]s <math> N </math> less than 1000 can be written as the sum of <math> j </math> consecutive [[positive integer | positive]] [[odd integer]]s for exactly 5 values of <math> j\ge 1 </math>?
+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]].
 == Solution ==

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

Revision as of 14:48, 25 September 2007

Problem

Solution

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