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We consider resonance problems for the one-dimensional p-Laplacian assuming Dirichlet boundary conditions. In particular, we consider resonance problems associated with the first three curves of the Fučík Spectrum. Using variational arguments based on linking theorems, we prove sufficient conditions for the existence of at least one solution. Our results are related to the classical Fredholm Alternative for linear operators. We also provide a new variational characterization for points on the third Fučík curve.