# Given $x,y,z\in \mathbb C$, $x+y+z=0$ and $|x|=|y|=|z|$, prove that $x^3=y^3=z^3$.

Given $$z_1,z_2,z_3\in \mathbb C$$ such that $$z_1+z_2+z_3=0 \tag {C1}$$ $$|z_1|=|z_2|=|z_3|, \tag{C2}$$ prove that $$z_1^3=z_2^3=z_3^3.\tag{C3}$$

Source: List of problems for math-contest training.

My attempt: By (C2), let $$\rho$$ be the common modulus for $$|z_j|,j=1,2,3.$$ Therefore $$\forall j$$, $$z_j=\rho e^{i\theta_j}\ \ \text{and} \ \ z_j^3=\rho^3 e^{i 3\theta_j}$$ And, by (C1) it holds that $$e^{i \theta_1}+e^{i \theta_2}+e^{i \theta_3}=0.$$ But (C3) implies $$e^{i 3\theta_1}=e^{i 3\theta_2}=e^{i 3\theta_3}.$$ and that implies $$\theta_1=\theta_2= \theta_3$$ (for $$\theta_j \in [0,2\pi[$$). But that appears to be inconsistent with (C1).

Is there a problem with the problem statement? hints and answers are welcomed.

(C3) doesn't imply that $$\theta_1=\theta_2=\theta_3$$.
Take for example $$\theta_1 = 0, \theta_2 = \frac{2\pi}{3}, \theta_3 = \frac{4\pi}{3}$$
Then $$e^{i3\theta_1} = e^0 = 1$$ $$e^{i3\theta_2} = e^{2\pi i} = 1$$ $$e^{i3\theta_3} = e^{4\pi i} = 1$$
But clearly $$\theta_1\not = \theta_2, \theta_1 \not = \theta_3,\theta_2\not = \theta_3$$.