MathDB

We are given a set

S=\{z_1,z_2,\cdots ,z_{1993}\}

, where

z_1,z_2,\cdots ,z_{1993}

are nonzero complex numbers (also viewed as nonzero vectors in the plane). Prove that we can divide

S

into some groups such that the following conditions are satisfied: (1) Each element in

S

belongs and only belongs to one group; (2) For any group

p

, if we use

T(p)

to denote the sum of all memebers in

p

, then for any memeber

z_i (1\le i \le 1993)

p

, the angle between

z_i

and

T(p)

does not exceed

90^{\circ}

; (3) For any two groups

p

and

q

, the angle between

T(p)

and

T(q)

exceeds

90^{\circ}

(use the notation introduced in (2)).

Problem Statement