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Romania District MO 2022 Grade 10 P2

Source: Romania District MO 2022 Grade 10

March 28, 2022
complex numbersromaniainequalities

Problem Statement

Let z1,z2z_1,z_2 and z3z_3 be complex numbers of modulus 1,1, such that ∣zi−zj∣≥2|z_i-z_j|\geq\sqrt{2} for all i≠j∈{1,2,3}.i\neq j\in\{1,2,3\}. Prove that ∣z1+z2∣+∣z2+z3∣+∣z3+z2∣≤3.|z_1+z_2|+|z_2+z_3|+|z_3+z_2|\leq 3.Mathematical Gazette
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