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Inequality "behind" complex numbers

Source: 2021 China Mathematical Olympiad, P1

November 24, 2020

inequalitiescomplex numbers

Problem Statement

Let

\{ z_n \}_{n \ge 1}

be a sequence of complex numbers, whose odd terms are real, even terms are purely imaginary, and for every positive integer

k

|z_k z_{k+1}|=2^k

. Denote

f_n=|z_1+z_2+\cdots+z_n|,

for

n=1,2,\cdots

(1) Find the minimum of

f_{2020}

. (2) Find the minimum of

f_{2020} \cdot f_{2021}