11. A5 (BLR) ${ }^{\mathrm{IMO3}}$ 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 of 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 $C$ such that $B C / A B=\lambda$.
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 $M$.
11. We claim that for $\lambda \geq \frac{1}{n-1}$ we can take all fleas as far to the right as we want. In every turn we choose the leftmost flea and let it jump over the rightmost one. Let $d$ and $\delta$ denote the maximal and the minimal distances between two fleas at some moment. Clearly, $d \geq(n-1) \delta$. After the leftmost flea jumps over the rightmost one, the minimal distance does not decrease, because $\lambda d \geq \delta$. However, the position of the leftmost flea moved to the right by at least $\delta$, and consequently we can move the fleas arbitrarily far to the right after a finite number of moves.
Suppose now that $\lambda0$, this proves the result.