Actually, none of you explained this as clearly as you could have.
The question is ill-formed since it doesn't tell you what the "speeds" are relative to. So the answer is really kind of just a wordplay if you will - not a physics issue.
Define speed of the belt and the plane
relative only to the ground. Belt goes backwards at 1 mph - simultaneously, the plane goes whatever speed it needs to relative to the belt in order to go 1 mph relative to the ground. It actually will be going 2 mph relative to the belt.
So, belt is going 1 mph aft relative to the ground and airplane is going 1 mph forward relative to the ground.
The plane's wheel speed will have to be twice their normal speed in order to take-off since at Vr = 160 kts (relative to the ground), belt will be going backwards at 160 kts and wheels will be going 320 kts.
The trick on this problem is to ask clearly, when it is first stated, what the speeds of each item are relative to. As originally formulted, it is an ill-formed question. If speed of plane and belt are relative to each other, the question is nonsensical.
in