  We present some cosmological solutions of Brans-Dicke theory, characterized
by a decaying vacuum energy density and by a constant relative matter density.
With these features, they shed light on the cosmological constant problems,
leading to a presently small vacuum term, and to a constant ratio between the
vacuum and matter energy densities. By fixing the only free parameter of our
solutions, we obtain cosmological parameters in accordance with observations of
the relative matter density, the universe age and redshift-distance relations.
