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Consider the thirty 30th roots of unity z = e^{2*pi*i*k/30} for k =…
Question
Consider the thirty 30th roots of unity z = e^{2*pi*i*k/30} for k = 0, 1, ..., 29. How many of them satisfy the strict inequality |z^2 + z + 1| < 1? Report the count.
✓ Verified answer: 14checked by our engine — not a guess
Step-by-step solution
Write z = e^{i*a} with a = 2*pi*k/30 = k*pi/15.
Then z^2 + z + 1 = e^{ia}(e^{-ia} + 1 + e^{ia}) = e^{ia}(1 + 2cos a).
So |z^2 + z + 1| = |1 + 2cos a|.
The condition |1 + 2cos a| < 1 means -1 < 1 + 2cos a < 1, i.e.
-1 < cos a < 0.
Thus a must lie strictly in (pi/2, 3*pi/2) excluding the point where cos a = -1 is not relevant (we only need cos a in (-1,0)).
With a = k*pi/15, cos a in (-1, 0) holds for k = 8, 9, ..., 22 except k = 15 (where cos a = -1 gives |1 + 2cos a| = 1, not < 1).
The integers 8..22 number 15, removing k = 15 leaves 14 values.
Final answer14
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