For Fredholm integral equations of the first kind with nondegenerate kernels the Picard criterion simultaneously provides an existence criterion and elucidates the essential ill-posedness of the problem. The author develops a discrete Picard condition for the overdetermined ill- conditioned linear algebraic systems which arise from the discretization of the first kind Fredholm equations. Essentially, the condition is that the Fourier coefficients of the right hand side, in terms of the generalized singular value decomposition associated with a regularized problem, decay to zero faster on average than the generalized singular values. The author proposes a numerical check of the satisfaction of the discrete Picard condition based on a moving geometric mean of the Fourier coefficients of the right hand side. Some numerical illustrations of the ideas as applied to Fredholm integral equations of the first kind are proved.