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Part of 1986 China National Olympiad

Problems(1)

China Mathematical Olympiad 1986 problem3

Source: China Mathematical Olympiad 1986 problem3

1/17/2014
Let Z1,Z2,⋯ ,ZnZ_1,Z_2,\cdots ,Z_n be complex numbers satisfying ∣Z1∣+∣Z2∣+⋯+∣Zn∣=1|Z_1|+|Z_2|+\cdots +|Z_n|=1. Show that there exist some among the nn complex numbers such that the modulus of the sum of these complex numbers is not less than 1/61/6.
complex numbersalgebra unsolvedalgebra