CMO 1993 Problem 4

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 $z_i (1\le i \le 1993)$ of $p$ , 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)).