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I'd like some assistance solving this complex number problem.
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Let z_1, z_2 be complex solutions of the equation az2 bz c = 0 (a,b,c in R), such that z_1, z_2 have a nonzero imaginary part and |2z_1 - 1/9| = |z_1 - z_2|.
Assume |z_1| = 1/sqrt(k). Let w be a solution of the equation cw2 bw a = 0.
How many integers k are there such that for each k, there are exactly nine complex numbers z_3 satisfying:
- z_3 has an integer imaginary part
- z_3 - w is a pure imaginary number (edit: 0 is considered a pure imaginary number, as 0 = 0i.)
- |z_3| ≤ |w|?
What would be the shortest way to solve this problem?
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