[hide] I calculate that it's impossible to run fast enough on the second lap to get your average speed up to twice the average speed of the first lap. For example, if you run the first lap in a minute (thus averaging one lap per minute), you would have to complete the second lap in no time at all in order for your total average speed to be two laps per minute. The same type of calculation shows that you always need to run infinitely fast on the second lap, regardless of your speed on the first lap.
A couple of observations: First, in real life it's impossible to measure speed with infinite precision, so it might be of some interest to calculate just how fast you would have to run in order to get your time on the second lap down to zero, when rounded to the precision of your stop watch. Second, I am not sure why you emphasized that the first lap has to be run at a "constant speed". It seems to me that you would still need to run the second lap instantaneously, regardless of how variable your speed was on the first lap.[/hide]