Problem 3

Let

n\ge2

be a positive integer. Initially there are

n

fleas on a horizontal line, not all at the same point. For a positive real number

\lambda

, a move chooses fleas at

A

and

B

with

A

to the left of

B

, and lets the flea at

A

jump to

C

to the right of

B

so that

BC=\lambda AB

. Determine all

\lambda

such that, for every point

M

and every initial position, a finite sequence of moves puts all fleas to the right of

M

Step 3 of 4: Iterate the greedy move

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In plain words

The rightmost point advances uniformly.

X\text{ increases or stays constant}

Detailed analysis

Each greedy move advances the current leftmost flea past the current rightmost one by at least the preserved gap. After cycling through the fleas, all positions have moved right by a positive amount; repeating makes the leftmost position unbounded.