AUSTRIAN-POLISH 1995 Problem 8

Consider the cube with the vertices at the points $(\pm 1, \pm 1, \pm 1)$ . Let $V_1,...,V_{95}$ be arbitrary points within this cube. Denote $v_i = \overrightarrow{OV_i}$ , where $O = (0,0,0)$ is the origin. Consider the $2^{95}$ vectors of the form $s_1v_1 + s_2v_2 +...+ s_{95}v_{95}$ , where $s_i = \pm 1$ . (a) If $d = 48$ , prove that among these vectors there is a vector $w = (a, b, c)$ such that $a^2 + b^2 + c^2 \le 48$ . (b) Find a smaller $d$ (the smaller, the better) with the same property.