If I go halfway to the town (which is 60 km away) at the speed of 30 km/hour, how fast do I have to go for the rest of the way to have the average speed of the entire way 60 km/hour?

The solution of not possible makes sense to me, and I understand the questions of the people above. I am going to try to answer those questions, but I'm bad at wording, so sorry if it doesn't help.

The question is asking for the average speed of the ENTIRE trip. Therefore, with a total distance of 120 km and an average speed of 60 km/hour, the average time you must take is 2 hours. Because you already took 2 hours to complete the first half, you are out of time.

Now the question is why can't you speed up to 120 km/hour for the second half and then average that to get 60 km/hour. Well, the thing is that when you average the two speeds to get the average speed for the entire trip, you come up with (30 + 120)/2=75 km/hour. As you can see, this is not 60 km/hour and does not satisfy the problem. You must be wondering why I didn't include the distances in that small equation. Well, it is asking for average SPEED, not average speed OVER A GIVEN DISTANCE. Because of that, you can only use the two speeds you went at to average 60.

I apologize for the last couple of sentences, because I kind of repeated myself. I hope that makes sense.