Mathegoras

Complex numbers › The complex plane › Extremes of the modulus

Minimum of a sum of |1+z^k|

High school · ●●●●○ 100 pts

Let the complex number zz satisfy ∣z∣=1|z|=1. Find the minimum possible value of 2021∣1+z∣+∣1+z2∣+∣1+z3∣+⋯+∣1+z4043∣2021|1+z|+|1+z^{2}|+|1+z^{3}|+\cdots+|1+z^{4043}|.

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triangle inequalityoptimizationroots of unity