Problem

Source: Iran MO Third Round A3

Tags: algebra, complex numbers, absolute value



Prove that for natural number $n$ it's possible to find complex numbers $\omega_1,\omega_2,\cdots,\omega_n$ on the unit circle that $$\left\lvert\sum_{j=1}^{n}\omega_j\right\rvert=\left\lvert\sum_{j=1}^{n}\omega_j^2\right\rvert=n-1$$iff $n=2$ occurs.