MathDB

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.

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