We are given a set S={z1,z2,⋯,z1993}, where z1,z2,⋯,z1993 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 zi(1≤i≤1993) of p, the angle between zi and T(p) does not exceed 90∘;
(3) For any two groups p and q, the angle between T(p) and T(q) exceeds 90∘ (use the notation introduced in (2)). vectorcomplex numberscombinatorics unsolvedcombinatorics