Sounds like a point in favor of going back to gender segegated schools. Let the girls cooperate, and the boys compete, and theoretically everyone will end off better for it.
Sooner or later they need to work together anyway.
I'd rather suggest being honest about the fact that statistically speaking boys and girls are different and both have to respect each other.
Anway, competition is normal in the working world (at least in America). So there's a problem regardless of school effects.