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$. | ||
− | + | == Solutions == | |
− | |||
− | == | ||
{{solution}} | {{solution}} | ||
− | == See | + | == See also == |
− | + | * <url>viewtopic.php?t=84558 Discussion on AoPS/MathLinks</url> | |
− | * | ||
− | |||
+ | {{USAMO newbox|year=2006|num-b=4|num-a=6}} | ||
[[Category:Olympiad Number Theory Problems]] | [[Category:Olympiad Number Theory Problems]] | ||
{{MAA Notice}} | {{MAA Notice}} |
Revision as of 03:12, 6 August 2014
Problem
(Zoran Sunik) A mathematical frog jumps along the number line. The frog starts at 1, and jumps according to the following rule: if the frog is at integer , then it can jump either to or to where is the largest power of 2 that is a factor of k\ge 2i2^i k2^i$.
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 | |
1 • 2 • 3 • 4 • 5 • 6 | ||
All USAMO Problems and Solutions |
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