IMO 1983 LL ROM53

Let a \inR and let z1, z2, . . . , zn be complex numbers of mod-

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IMO 1983 LL ROM53

Origin: ROM

Problem

Let a \inR and let z1, z2, . . . , zn be complex numbers of mod- ulus 1 satisfying the relation n  k=1 z3 k = 4(a + (a −n)i) −3 n  k=1 zk. Prove that a \in{0, 1, . . ., n} and zk \in{1, i} for all k.