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 < | + | (''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 02:15, 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 
, then it can jump either to 
or to 
where 
is the largest power of 2 that is a factor of . Show that if 
is a positive integer and 
is a nonnegative integer, then the minimum number of jumps needed to reach 
is greater than the minimum number of jumps needed to reach 
.
Solutions
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See also
- <url>viewtopic.php?t=84558 Discussion on AoPS/MathLinks</url>
| 2006 USAMO (Problems • Resources) | ||
| Preceded by Problem 4 |
Followed by Problem 6 | |
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| All USAMO Problems and Solutions | ||
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