Estimates of the Martian elastic lithosphere thickness suggest small values of similar to 25 km during the Noachian for the southern hemisphere and a large present-day difference below the two polar caps (\textgreater= 300 km in the north and \textgreater110 km in the south). In addition, young lava flows suggest that Mars has been volcanically active up to the recent past. We run Monte Carlo simulations using a 1-D parameterized thermal evolution model to investigate whether a north/south hemispheric dichotomy in crustal properties and composition can explain these constraints. Our results suggest that 55-65% of the bulk radioelement content are in the crust, and most of it (43-51%) in the southern one. The southern crust can be up to 480 kg/m(3) less dense than the northern one and might contain a nonnegligible proportion of felsic rocks. Our models predict a dry mantle and a wet or dry crustal rheology today. This is consistent with a mantle depleted in radioelements and volatiles. We retrieve north/south surface heat flux of 17.1-19.5 mW/m(2) and 24.8-26.5 mW/m(2), respectively, and a large difference in lithospheric temperatures between the two hemispheres (170-304 K in the shallow mantle). This difference could leave a signature in the seismic signals measured by the future InSight mission.