3. A flea jumps along the number line, and the length of each jump cannot be less than $n$. It starts its movement from the origin and wants to visit all integer points belonging to the segment $[0,2013]$ (and only them!) exactly once. For what greatest value of $n$ will it be able to do this?
Answer: $n=1006$.

Solution: For $n=1006$, a path can be constructed as follows:

$$
0 \rightarrow 1007 \rightarrow 1 \rightarrow 1008 \rightarrow \ldots \rightarrow 1005 \rightarrow 2012 \rightarrow 1006 \rightarrow 2013
$$

We will prove that $n$ cannot be greater than 1006. Indeed, suppose $n \geqslant 1007$. Then the point with coordinate 1007 can only be reached from the start of the segment (point 0). However, if the flea jumps from there to point 1007, it cannot jump back, so it must end its path at this point and will not visit other points on the segment.