Fulton asked: `How many solutions to a problem of enumerative geometry
can be real?'.
In this paper, we consider problems of enumerating p-planes
having excess intersection with general linear subspaces and show that
there is a choice of real linear subspaces osculating the rational normal
curve so that all p-planes having excess intersection are real.
This proves a special case of the conjecture of Shapiro and Shapiro.