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

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== Problem ==
 
== 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 <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$.
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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</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>.
  
 
== Solutions ==
 
== Solutions ==

Revision as of 03:15, 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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