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 4 of 4: State the answer

In plain words

Reach every target point.

\boxed{\lambda\ge\frac1{n-1}}

Detailed analysis

Thus the admissible values are exactly $\lambda\ge1/(n-1)$ .