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Romanian District Olympiad 2019 - Grade 10 - Problem 2

Source: Romanian District Olympiad 2019 - Grade 10 - Problem 2

March 17, 2019
complex numbersmodulusalgebra

Problem Statement

Let n∈N,n≥3.n \in \mathbb{N}, n \ge 3. a)a) Prove that there exist z1,z2,…,zn∈Cz_1,z_2,…,z_n \in \mathbb{C} such that z1z2+z2z3+…+zn−1zn+znz1=ni.\frac{z_1}{z_2}+ \frac{z_2}{z_3}+…+ \frac{z_{n-1}}{z_n}+ \frac{z_n}{z_1}=n \mathrm{i}. b)b) Which are the values of nn for which there exist the complex numbers z1,z2,…,zn,z_1,z_2,…,z_n, of the same modulus, such that z1z2+z2z3+…+zn−1zn+znz1=ni?\frac{z_1}{z_2}+ \frac{z_2}{z_3}+…+ \frac{z_{n-1}}{z_n}+ \frac{z_n}{z_1}=n \mathrm{i}?
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