Difference between revisions of "2000 IMO Problems/Problem 3"
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==Problem== | ==Problem== | ||
| − | Let <math>n \ge 2</math> be a positive integer and <math>\lambda</math> a positive real number. | + | 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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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>. | ||
Revision as of 00:12, 19 November 2023
Problem
Let 
be a positive integer and 
a positive real number. Initially there are 
fleas on a horizontal line, not all at the same point. We define a move as choosing two fleas at some points 
and 
to the left of , and letting the flea from jump over the flea from to the point 
so that 
.
Determine all values of such that, for any point 
on the line and for any initial position of the fleas, there exists a sequence of moves that will take them all to the position right of .
Solution
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See Also
| 2000 IMO (Problems) • Resources | ||
| Preceded by Problem 2 |
1 • 2 • 3 • 4 • 5 • 6 | Followed by Problem 3 |
| All IMO Problems and Solutions | ||
