There is more than one way to solve this. First, let's look at the easy way. The two cars were 10 km apart. They crash halfway between them in half an hour. The bird travels 10 km in that half an hour. That's the answer.

How about the hard way? The bird goes twice as fast as either car. So the bird initially covers 20/3 km while the car A covers 10/3 km. Why?  V*T = D.  10*t + 20*t = 10.  Thus, t = 10/30, or t = 1/3.  Car A goes (10)*(1/3) and the bird goes (20)*(1/3) and so on…

The bird turns around. Car B is now 10/3 km away. The bird returns to car B, going twice as fast as car B. So the fly goes 20/9 km while car B goes 10/9 km. Then the bird turns around and goes 20/27 km, while car A goes 10/27 km. After an infinite number of flights, the bird has gone 20/3+20/9+20/27+20/81+... That appears to be a Geometric Series. The sum is S=a/(1-r), where a is the first term 20/3, and r is the ratio of successive terms 1/3. So the sum is 10 km.