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

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== Problem ==
 
== Problem ==
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(''Zoran Sunik'') 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>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 2 that is a factor of <math>n . 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^i k<math> is greater than the minimum number of jumps needed to reach </math>2^i$.
  
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>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 2 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^i k </math> is greater than the minimum number of jumps needed to reach <math>2^i </math>.
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== Solutions ==
 
 
== Solution ==
 
  
 
{{solution}}
 
{{solution}}
  
== See Also ==
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== See also ==
 
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* <url>viewtopic.php?t=84558 Discussion on AoPS/MathLinks</url>
* [[2006 USAMO Problems]]
 
* [http://www.artofproblemsolving.com/Forum/viewtopic.php?p=490682#p490682 Discussion on AoPS/MathLinks]
 
  
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{{USAMO newbox|year=2006|num-b=4|num-a=6}}
  
 
[[Category:Olympiad Number Theory Problems]]
 
[[Category:Olympiad Number Theory Problems]]
 
{{MAA Notice}}
 
{{MAA Notice}}

Revision as of 03:12, 6 August 2014

Problem

(Zoran Sunik) 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 $n$, then it can jump either to $n+1$ or to $n+2^{m_n+1}$ where $2^{m_n}$ is the largest power of 2 that is a factor of $n . Show that if$k\ge 2$is a positive integer and$i$is a nonnegative integer, then the minimum number of jumps needed to reach$2^i k$is greater than the minimum number of jumps needed to reach$2^i$.

Solutions

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See also

  • <url>viewtopic.php?t=84558 Discussion on AoPS/MathLinks</url>
2006 USAMO (ProblemsResources)
Preceded by
Problem 4
Followed by
Problem 6
1 2 3 4 5 6
All USAMO Problems and Solutions

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