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>
 
== Solution ==
 
== Solution ==
 
== See Also ==
 
== See Also ==
 
*[[2006 USAMO Problems]]
 
*[[2006 USAMO Problems]]

Revision as of 11:08, 12 July 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 $n$, 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 $2^ik$ is greater than the minimum number of jumps needed to reach $2^i.$

Solution

See Also

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