If I'm not totally dumb the car would move when the right magnet would not be attached to the car, at same distance of the booth magnets.
So why would it not move?
A simple symmetry argument might help here. Try to abstract everything away from your picture that is just a distraction. In the end it might look like this:
Why should this move to the left or to the right? It cannot and therefore adding a person and wheels does not change the problem.
The left magnet is pulled forward, pulling the car behind it.
The right magnet is pulled backward, pushing the car ahead of it via the rod that attaches it to the car. (The fact that the rod loops over the car and attaches to the back makes no difference.)
If the rod is flexible, the car will move forward as the magnets move closer together, but only until the magnets meet or the rod reaches the limit of its flexibility.
With careful control of the movement of the magnets (or any weight, magnetic or not), you can achieve some net forward motion by taking advantage of friction, since friction is non-linear. A slow movement won't overcome static friction, and the car doesn't move; a quick jerk in the other direction will cause the car to move. This doesn't violate any conservation law; the car is effectively pushing against the surface it's sitting on, just like any ordinary wheeled vehicle. It won't work on a frictionless surface.
If the forward magnet isn't attached to the car at all, the car will be pulled forward until the magnets meet. If the forward magnet is kept at a constant distance from the car as the car moves, you've got a propulsion system. But again, no conservation laws are violated; you still have to have some way to move the forward magnet.
But the original scheme would probably work in a universe that follows Warner Brothers cartoon physics. Working out a consistent mathematical model for that is left as an exercise.
Thank you for your interest in this question. Because it has attracted low-quality answers, posting an answer now requires 10 reputation on this site.
Would you like to answer one of these unanswered questions instead?