Difference between revisions of "2006 USAMO Problems/Problem 5"

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== Problem ==
 
== Problem ==
A mathematical frog jumps along the number line. The frog starts at <math>1</math>, and jumps according to the following rule: if the frog is at integer <math>n</math>, then it can jump either to <math>n+1</math> or to <math>n+2^{m_n+1}</math> where <math>2^{m_n}</math> is the largest power of <math>2</math> that is a factor of <math>n</math>. Show that if <math>k\ge 2</math> is a positive integer and <math>i</math> is a nonnegative integer, then the minimum number of jumps needed to reach <math>2^ik</math> is greater than the minimum number of jumps needed to reach <math> 2^i.</math>
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A mathematical frog jumps along the number line. The frog starts at 1, and jumps according to the following rule: if the frog is at integer <math> \displaystyle n </math>, then it can jump either to <math> \displaystyle n+1 </math> or to <math>n+2^{m_n+1}</math> where <math>2^{m_n}</math> is the largest power of 2 that is a factor of <math> \displaystyle n </math>. Show that if <math>k\ge 2</math> is a positive integer and <math> \displaystyle i </math> is a nonnegative integer, then the minimum number of jumps needed to reach <math> \displaystyle 2^i k </math> is greater than the minimum number of jumps needed to reach <math> \displaystyle 2^i </math>.
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== Solution ==
 
== Solution ==
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{{solution}}
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== See Also ==
 
== See Also ==
*[[2006 USAMO Problems]]
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* [[2006 USAMO Problems]]
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* [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=490682#p490682 Discussion on AoPS/MathLinks]
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[[Category:Olympiad Number Theory Problems]]

Revision as of 19:48, 1 September 2006

Problem

A mathematical frog jumps along the number line. The frog starts at 1, and jumps according to the following rule: if the frog is at integer $\displaystyle n$, then it can jump either to $\displaystyle n+1$ or to $n+2^{m_n+1}$ where $2^{m_n}$ is the largest power of 2 that is a factor of $\displaystyle n$. Show that if $k\ge 2$ is a positive integer and $\displaystyle i$ is a nonnegative integer, then the minimum number of jumps needed to reach $\displaystyle 2^i k$ is greater than the minimum number of jumps needed to reach $\displaystyle 2^i$.

Solution

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See Also

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