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Omni-MATH / Let n≥2 be a positive integer and λ 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 s…

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problem

Let n≥2 n \geq 2 be a positive integer and λ \lambda a positive real number. Initially there are n n fleas on a horizontal line, not all at the same point. We define a move as choosing two fleas at some points A A and B B, with A A to the left of B B, and letting the flea from A A jump over the flea from B B to the point C C so that $ \frac {BC}{AB} \equal{} \lambda$. Determine all values of λ \lambda such that, for any point M M on the line and for any initial position of the n n fleas, there exists a sequence of moves that will take them all to the position right of M M.
Plain-text mathematical notation (without MathML)
Let n≥2 be a positive integer and λ 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, with A 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} \equal{} \lambda$. 

Determine all values of λ 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.
Original LaTeX notation
Let $ n \geq 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$, with $ A$ 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} \equal{} \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$.

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task_id
1084e251-a4a1-57e2-9419-dc0a686760d3
task_key
test--1084e251-a4a1-57e2-9419-dc0a686760d3
task_revision_id
1
config / split
/ test

Official source

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