  The Beauville-Voisin conjecture for a hyperk\"ahler manifold X states that
the subring of the Chow ring A^*(X) generated by divisor classes and Chern
characters of the tangent bundle injects into the cohomology ring of X. We
prove a weak version of this conjecture when X is the Hilbert scheme of points
on a K3 surface, for the subring generated by divisor classes and tautological
classes. This in particular implies the weak splitting conjecture of Beauville
for these geometries. In the process, we extend Lehn's formula and the
Li-Qin-Wang W_{1+infinity} algebra action from cohomology to Chow groups, for
the Hilbert scheme of an arbitrary smooth projective surface S
