We report the observation of a purely entropic demixing transition in a three-dimensional binary hard-core mixture by computer simulations. This transition is observed in a lattice model of a binary hard-core mixture of parallel cubes provided that the size asymmetry of the large and small particles is sufficiently large (greater than or equal to 3, in the present case). In addition, we have performed simulations of a single athermal polymer in a hard-core solvent. As we increase the chemical potential of the solvent, we observe a purely entropy-driven collapse of the polymer : the scaling of the radius of gyration R(g) of the polymer with the number of segments N changes from that of a polymer in a good solvent to that of a collapsed polymer. Both for the study of the hard-core demixing and of the polymer collapse, it was essential to use novel collective Monte Carlo moves to speed up equilibration. We show that in the limit sigma(1)/sigma(2)-->0, the pair distribution function for an off-lattice binary hard-core mixture of parallel cubes with side lengths sigma(1) and sigma(2) diverges at contact for the large particles. For the lattice system, we calculated the pair distribution functions g(r) up to the fourth virial coefficient. The difference in g(r) at contact for a binary system and a pure system at the same packing fraction gives a rough criterion, whether the mixture phase separates.