Given a general plane curve Y of degree d, we compute the number n_d ofirreducible plane conics that are 5-fold tangent to Y. This problem has beenstudied before by Vainsencher using classical methods, but it could not besolved there because the calculations received too many non-enumerativecorrection terms that could not be analyzed. In our current approach, weexpress the number n_d in terms of relative Gromov-Witten invariants that canthen be directly computed. As an application, we consider the K3 surface givenas the double cover of P^2 branched along a sextic curve. We compute the numberof rational curves in this K3 surface in the homology class that is thepull-back of conics in P^2, and compare this number to the correspondingYau-Zaslow K3 invariant. This gives an example of such a K3 invariant for anon-primitive homology class.
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