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Difference between revisions of "2000 IMO Problems/Problem 3"

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==Problem==
 
==Problem==
  
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Let <math>n \ge 2</math> be a positive integer and <math>\lambda</math> a positive real number.  
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Let <math>n \ge 2</math> be a positive integer and <math>\lambda</math> a positive real number. Initially there are <math>n</math> fleas on a horizontal line, not all at the same point. We define a move as choosing two fleas at some points <math>A</math> and <math>B</math> to the left of <math>B</math>, and letting the flea from <math>A</math> jump over the flea from <math>B</math> to the point <math>C</math> so that <math>\frac{BC}{AB}=\lambda</math>.
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Initially there are <math>n</math> fleas on a horizontal line, not all at the same point. We define a move as choosing two fleas at some points <math>A</math> and <math>B</math> to the left of <math>B</math>, and letting the flea from <math>A</math> jump over the flea from <math>B</math> to the point <math>C</math> so that <math>\frac{BC}{AB}=\lambda</math>.
 
  
 
Determine all values of <math>\lambda</math> such that, for any point <math>M</math> on the line and for any initial position of the <math>n</math> fleas, there exists a sequence of moves that will take them all to the position right of <math>M</math>.
 
Determine all values of <math>\lambda</math> such that, for any point <math>M</math> on the line and for any initial position of the <math>n</math> fleas, there exists a sequence of moves that will take them all to the position right of <math>M</math>.
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==See Also==
 
==See Also==
  
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{{IMO box|year=2000|num-b=2|num-a=3}}
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{{IMO box|year=2000|num-b=2|num-a=4}}

Latest revision as of 23:16, 18 November 2023

Problem

Let $n \ge 2$ be a positive integer and $\lambda$ a positive real number. Initially there are $n$ fleas on a horizontal line, not all at the same point. We define a move as choosing two fleas at some points $A$ and $B$ to the left of $B$, and letting the flea from $A$ jump over the flea from $B$ to the point $C$ so that $\frac{BC}{AB}=\lambda$.

Determine all values of $\lambda$ such that, for any point $M$ on the line and for any initial position of the $n$ fleas, there exists a sequence of moves that will take them all to the position right of $M$.

Solution

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See Also

2000 IMO (Problems) • Resources
Preceded by
Problem 2 		1 • 2 • 3 • 4 • 5 • 6 Followed by
Problem 4
All IMO Problems and Solutions
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