Your first impulse might be to give the answer as
500 mph, the aircraft’s own airspeed and the arithmetic
mean of 450 and 550. But first impulses can be
wrong. Let the distance between the cities be d and
the required speed, v. The time it takes the aircraft
to complete the outbound stretch (flying against
the headwind) is d/450, whereas the time of the
return stretch is d/550. The total time is, therefore,
d/450 + d/550. Setting this equal to the time it would
take to complete the round trip at a constant ground
speed v, we have