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Hard inequality involving complex numbers from Saint Petersburg

Source: Open math olympiad of lyceum 239

April 8, 2021
complex numbersinequalities

Problem Statement

Let a,b,ca,b,c be some complex numbers. Prove that ∣a2ab+ac−bc∣+∣b2ba+bc−ac∣+∣c2ca+cb−ab∣≥32|\dfrac{a^2}{ab+ac-bc}| + |\dfrac{b^2}{ba+bc-ac}| + |\dfrac{c^2}{ca+cb-ab}| \ge \dfrac{3}{2} if the denominators are not 0
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