A bumper system (4) for a motor vehicle comprising a cross beam (1) which is globally orientated in a transverse direction (Y), said cross beam comprising a front wall (6) adapted to receive a crash impact force and a rear wall (5) opposed and spaced from said front wall, at least one absorber (2), at least an intermediate component (3) to attach the absorber (2) to the cross beam (1), said intermediate component (3) being connected to the rear wall (5) through a first contact area (10), distant from a longitudinal axis (LL), by an internal component distance, called Lin, and an external component distance, called Lout, which corresponds respectively to the minimum and maximum distance between said first contact area (10) and the longitudinal axis (LL), said longitudinal axis (LL) passing at mid width of the cross beam and being perpendicular to the transverse direction (Y), said intermediate component (3) being connected to the absorber (2) through a second contact area (20) distant from the longitudinal axis (LL) by an internal profile distance, called Din, and an external profile distance, called Dout, which corresponds respectively to the minimum and maximum distance between the second contact (20) area and the longitudinal axis (LL), wherein a part of the intermediate component distant from the longitudinal axis (LL) by a distance comprised between Lout and Dout has a smaller bending stiffness (Sinterm) about a vertical axis (Z) than a part of the cross beam distant from the longitudinal axis (LL) by a distance comprised between Lout and Din (Scross), said vertical axis (Z) being perpendicular to the transverse direction (Y) and the longitudinal axis (LL).