Problem 1878

Let $n \geq 2$ be a positive integer and $\lambda$ a positive real number. There are $n$ fleas on a horizontal line, and we need to find the values of $\lambda$ for which, given any point $M$ and any initial positions of the fleas, there is a sequence of moves that can place all fleas to the right of $M$.

### Move Description:

A move consists of selecting two fleas located at points $A$ and $B$ (with $A$ to the left of $B$), and moving the flea from $A$ to a new point $C$ such that $\frac{BC}{AB} = \lambda$.

### Analysis:

- Assume the leftmost flea is initially at position $x_1$ and the rightmost flea is at position $x_n$.
- The goal is to transform the system such that all fleas are located at some position greater than $M$.

### Considerations:

1. Move Effect:
- If a flea initially at $A$ jumps to $C$, then:
$$C = A + \lambda(B - A) = (1 - \lambda)A + \lambda B.$$
- This replaces $A$ with a point closer to $B$ (if $\lambda > 0$).

2. Bounding Fleas to the Right:
- We need each flea to eventually move past $M$. Since fleas consecutively jump rightward, the greatest possible accumulation of fleas past $M$ occurs when effective $\lambda$ allows maximal stretching of intervals.

3. **Condition on $\lambda$:**
- Starting with fleas positioned in a finite interval covering $x_1$ to $x_n$, progressively applying transformations:
- If $\lambda$ is too small, the rightward jumps might be insufficient to clear $M$ in finite steps.

4. Sufficient Condition:
- Sufficiently large $\lambda$ ensures that the accumulative forward motion possible among successive intervals exceeds the necessary coverage over distance $x_n - M$.
- Analyzing proportion:
- For $m$ iterations to push gaps from $x_1$ through to beyond $x_n$, having $\lambda \ge \frac{1}{n-1}$ guarantees accumulative growth beyond necessary jumps.

### Conclusion:

With the above reasoning, we conclude that the values of $\lambda$ that ensure an eventual placement of all fleas to the right of any point $M$, for any initial configuration of fleas, are:

$$\boxed{\lambda \ge \frac{1}{n-1}}$$

This bound arises from ensuring that progressive cumulative extensions with each move can bridge the intervals ensuring encompassment reaches past any arbitrary point $M$.