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Prove these 3 properties

Source: 2019 Jozsef Wildt International Math Competition-W. 39

May 19, 2020
complex numbersnumber theory

Problem Statement

Let uu, vv, ww complex numbers such that: u+v+w=1u + v + w = 1, u2+v2+w2=3u^2 + v^2 + w^2 = 3, uvw=1uvw = 1. Prove that
[*] uu, vv, ww are distinct numbers two by two [*] If S(k)=uk+vk+wkS(k)= u^k + v^k + w^k, then S(k)S(k) is an odd natural number [*] The expressionu2n+1−v2n+1u−v+v2n+1−w2n+1v−w+w2n+1−u2n+1w−u\frac{u^{2n+1} - v^{2n+1}}{u-v}+\frac{v^{2n+1}-w^{2n+1}}{v-w}+\frac{w^{2n+1}-u^{2n+1}}{w-u}is an integer number.
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