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 2 of 4: Prove sufficiency at and above the threshold

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In plain words

Preserve the minimum gap.

\lambda\ge\frac1{n-1}

Detailed analysis

Now assume $\lambda\ge1/(n-1)$ . Let $d$ be the smallest gap and $D=x_n-x_1$ . Since $D$ is the sum of $n-1$ gaps, $D\ge(n-1)d$ , so $\lambda D\ge d$ . Move the leftmost flea over the rightmost one; the new minimum gap is at least $d$ .