Let t1 be the time taken to drive the first half of the route.

Let t2 be the time taken to drive the second half.

Let d1 be the distance traveled in the first half.

Let d2 be the distance traveled in the second half.

Let x be what we want to know (namely, the average speed during the second half of the route).

Then the following relations hold:

40 * (t1 + t2) = d1 * d2.

20 * t1 = d1.

x * t2 = d2.

d1 = d2.

Use algebra to solve for x.

To average 40 mph requires completing the trip in a certain amount of time, and even without doing any algebra, I notice that you will have used all of the available time just completing the first half of the trip, so you're speed would have to be infinitely fast during the second half.

I am pretty confident in that conclusion, but a little algebra will increase my confidence, so let us calculate as follows: the time you have to do the trip = t1 + t2 = d1 / 40 + d2 / 40, which (since d1 = d2) equals d1 / 20, but (by equation 2) d1 / 20 equals t1, so t2 must be zero.