KöMaL 2007 B4010

$p$($x$) and $q$($x$) are polynomials of real coefficients that have no real roots in common, and $p$ ( $q$($x$) ) =$q$ ( $p$($x$) ) for all $x$. Prove that the polynomials $p$ ( $p$($x$) ) and $q$ ( $q$($x$) ) have no common real roots either.