Difference between revisions of "2000 IMO Problems/Problem 3"
(Created page with "Let <math> n \geq 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 t...") |
(→See Also) |
||
(5 intermediate revisions by 2 users not shown) | |||
Line 1: | Line 1: | ||
− | + | ==Problem== | |
− | 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>. | + | 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>. |
+ | |||
+ | 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>. | ||
==Solution== | ==Solution== | ||
− | + | {{solution}} | |
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | |||
− | + | ==See Also== | |
− | + | {{IMO box|year=2000|num-b=2|num-a=4}} |
Latest revision as of 00:16, 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
This problem needs a solution. If you have a solution for it, please help us out by adding it.
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 |