This has stumped people on other boards and even started an argument on 4chan's IRC channel.
Imagine a jet plane (although it could probably be any sort of plane), and a runway that's a giant treadmill. When the plane's on the treadmill, the treadmill matches the plane's airspeed (relative to the ground) but in the opposite direction. So when the plane's at the end of the runway/treadmill going 0mph, the treadmill's going 0mph. When the plane's traveling at 80 knots facing north, the treadmill's going 80 knots southward.
Does the plane take off? This answer doesn't require complex math, just common sense.
I believe it can take off. I'll say why after a while, to see if someone else can explain it.