Answer: a fly and two bicycles (idea)

gorgonzola (writing below, but actually before; don't ask, it's a long story) gives the right answer, but works fairly hard for it. So let's start with the answer: the fly could be anywhere, or, in other words, the problem is ill posed and has no answer. The true math geeks among you will see that the proposal to take δ->0 doesn't work (the starting point doesn't converge as δ->0, unfortunately).

The time-reversed case is just what's described in old chestnut: two bicycles and a fly, for varying initial positions of the fly.

So how can we see this immediately? Well, as gorgonzola does, we should reverse time. But we can cut through the icky mathematics and do it in one fell swoop. Reversing time, we know the bicycles start 10km from the centre point. So let's pick any starting (ending) position between them for the fly, and let time flow backwards. After 1 hour, the bicycles crash together (start out), and the fly has to be between them. So the fly too is at the centre position. But this is true no matter which starting position we chose for the fly!

We see that in the reversed case, information (the fly's starting/ending position) gets destroyed. Since information cannot get created, there is no solution, as gorgonzola correctly states.