This paper is about the gap in the mathematical skills of high school students between North and South of Italy. In particular we investigate the success in Italian mathematical olympiads and in the contest for the fellowships given by the Istituto Nazionale di Alta Matematica to students enrolling in the Corso di Laurea in matematica. Against to what might have been expected, we show that even in these very high level skills there is the same (or even a bigger) gap as the one which has been pointed out by the OECD-PISA study for 15-years old students. We argue that a suitable policy of rewards to (not necessarily extremely) skilled students and to their teachers, is needed. In particular, a system of voluntary math skill certification could be useful.