  Given a general plane curve Y of degree d, we compute the number n_d of
irreducible plane conics that are 5-fold tangent to Y. This problem has been
studied before by Vainsencher using classical methods, but it could not be
solved there because the calculations received too many non-enumerative
correction terms that could not be analyzed. In our current approach, we
express the number n_d in terms of relative Gromov-Witten invariants that can
then be directly computed. As an application, we consider the K3 surface given
as the double cover of P^2 branched along a sextic curve. We compute the number
of rational curves in this K3 surface in the homology class that is the
pull-back of conics in P^2, and compare this number to the corresponding
Yau-Zaslow K3 invariant. This gives an example of such a K3 invariant for a
non-primitive homology class.
