341yes and that is reminiscent of the technique used in proving Fermat's Little Theorem.
Let S = (z_1,z_2,...,z_{2010}) We construct for each z_i \in S the set S_i = (z_iz_1,z_i,z_2,...,z_iz_{2010})
If it is true that S_i = S for 1 \le i \le 2010 then prove that
(1) |z_i| =1
(2) z \in S \Rightarrow \overline{z} \in S
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2 Answers
66Since Si = S, so the product of the elements in the two sets must be the same. That is
z1 z2 ... z2010 = zi 2010 z1 z2 ... z2010
which means that zi must be one of the 2010-th roots of unity. As such |zi| =1 for each 1 ≤ i ≤ 2010.
This incidentally also means that S is the set of the 2010-th roots of unity and the (2) one follows.
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