Good lord. I"ll agree that the question is not worded very well. But the people arguing that it won"t fly are having some serious problems understanding what is going on here.
First: let"s clarify the whole "conveyer belt moving at opposite the speed of the airplane" statement. The only way this makes sense is to assume this means the conveyer belt moves at the exact speed that the plane would be moving at were it on a regular, "immobile" runway, given it"s thrust. Thinking of it as instantaneously matching it"s speed relative to the surroundings is a complete paradox. Think about it, such a contraption could never even get started. Consider this situation: the conveyer belt has managed to slow the plane to a halt (ie not moving relative to the surroundings)... well, doesn"t this mean that the conveyer belt (as well as the plane) are both moving at 0 mph? It"s nonsense, and obviously not what the question intended.
Second: let"s get the whole weight/friction thing out of the way. Think of the airplane sitting still, with zero thrust being applied by the engine/prop. If the conveyer belt started at this point, with no thrust being applied by the engine/prop, then yes, the plane would move backwards with the conveyer belt. This is because the plane is not generating enough thrust to overcome the friction of the runway-tire-plane connection. This is similar to the "if you dropped the airplane onto a conveyer belt moving 100 mph" statement.
However the people arguing that the plane won"t take off like to make this out to be some monumental force to be overcome. It isn"t. It is TRIVIAL for an airplane to generate enough force to break through this barrier. You can go to an airport and see it done all the time, it"s called "going from 0 mph to >0 mph". If it wasn"t trivial, think about how much energy/fuel/etc it would take to get a plane to accelerate to ~200 kph. The force needed to overcome the friction barrier between the wheels and the ground is like a whimpering fart compared to the amount of thrust the engines/prop routinely generate to move the airplane fast enough to fly through the air.
Now if you understood all that, and still don"t see why the plane would take off, you really are an idiot.
Let"s run through the whole situation (this is mostly a rehash of some good posts in the physics thread link btw)...
Initial conditions: Plane is generating 0 mph worth of thrust (yes I know mph isn"t really a measure of thrust but I think you know what I mean). Conveyer belt is spinning at 0 mph. Plane is stationary with regards to the surroundings.
Now let"s assume that the engines are turned on very gradually, such that if the plane was on a normal runway, it would be stationary for a split second before beginning to move. Say it takes 1 "unit" of thrust to push the plane hard enough to break the bond of friction between runway and wheel.
Plane is generating 0.5 units of thrust. Conveyer belt is spinning with -0.5 units of thrust. Plane is moving BACKWARDS at some very miniscule speed.
The thrust increases to exactly 1 unit of thrust.
Plane is generating 1 unit of thrust. Conveyer belt is spinning with 1 unit of thrust. Plane is STATIONARY. AT THIS POINT AND ABOVE THE WHEELS CAN BE TREATED AS FREELY SPINNING.
Now the thrust goes above 1 unit. The thrust of the conveyer belt goes above 1 unit. If this were a car, with powered wheels, it would remain stationary with regards to the surroundings. A car is moved BY it"s wheels. The wheels turn, and the friction between them and the road moves the car forward. We"re not talking about a car. We"re talking about an airplane using a jet engine or a prop. The engine/prop sucks in air from the front and spits it out the back. With the (negligible) friction barrier broken, the runway and the wheel don"t matter AT ALL. The plane will accelerate porportional to how fast the engine/prop pulls it through the air, and eventually move through the air fast enough to generate lift on the wings.
A good analogy ripped from the physics forum link...
Instead of a plane on the conveyer belt, its you, you"re wearing rollerblades, and you have a (forward firing) jetpack on your back. You will move forward. This is literally the EXACT SAME situation as far as physics is concerned.
It feels really redundant writing out a post this long after so much has been said, but some people are just so wrong I can"t stand it.