A friend of mine introduced me to the following question: Does there exist a smooth function $f: \mathbb{R} \to \mathbb{R}$ ($f \in C^{\infty}$), such that $f$ maps rationals to rationals and irrationals to irrationals and is nonlinear?
He has been able to prove that such a polynomial (with degree at least 2) doesn't exist.
The problem has been asked before at least at http://www.artofproblemsolving.com.