For potential flow of unbounded incompressible perfect fluid d’Alembert proved that The solid body moving with constant velocity meets zero drag force. We discuss general conditions under which the paradox takes place, and examine some moving-contact problems of elasticity. The steady-state solution for an elastic half-plane under a moving indenter is derived based on the corresponding transient problem and on a condition concerning energy fluxes. The resulting stresses and displacements are found explicitly starting from their expressions in terms of a single analytical function. This solution incorporates all speed ranges, including the super-Rayleigh subsonic and intersonic speed regimes, which received no final description to date. Next, the wedging of an elastic plane is considered for a finite wedge moving at a distance of the crack tip. Finally, we solve the problem for a wedge moving along the interface of two elastic half-planes compressed together. It is found that in addition to the sub-Rayleigh speed regime, where the paradox takes place for a smooth frictionless indenter of arbitrary shape, there exists a sharp decrease in the resistance in a vicinity of the longitudinal wave speed with zero limit at this speed. For other speed regimes we determine the driving forces caused by the main underling factors: stress field singular points on the contact area (the super-Rayleigh subsonic speed regime), the wave radiation (intersonic and supersonic regimes) and the fracture resistance (the wedging problem).