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Geometry in algebra exam

Source: Iran MO 2023 3rd round , Algebra exam P1

August 21, 2023

geometrycomplex numbers

Problem Statement

Given

12

complex numbers

z_1,...,z_{12}

st for each

1 \leq i \leq 12

|z_i|=2 , |z_i - z_{i+1}| \geq 1

prove that :

\sum_{1 \leq i \leq 12} \frac{1}{|z_i\overline{z_{i+1}}+1|^2} \geq \frac{1}{2}