<s>[INST] Find all the real numbers an expression might take when $a, b, c$ are complex numbers of the same modulus. 
Let $a, b, c$ be three complex numbers of the same modulus. Find all real numbers that might be equal to:
$$x = \frac{a^3+b^3+c^3}{abc}$$

It is obvious that when all three numbers are equal, we might write that:
$$x = \frac{3a^3}{a^3} = 3 \in \mathbb{R}$$
I would write, in order to obtain real values:
$$x = \overline{x}$$
So:
$$\overline{a}\overline{b}\overline{c} (a^3 + b^3 + c^3) = abc (\overline{a}^3 + \overline{b}^3 + \overline{c}^3)$$
But the calculations are quite harsh, the only thing I might use to solve them is that:
$$a\overline{a} = b\overline{b} = c\overline{c} = m^2$$
where $m$ is the complex modulus of the three numbers.
 [/INST] Let us implement Mark Bennet's idea.
Let $\frac{a^3}{abc}=e^{i\alpha}$ and $\frac{b^3}{abc}=e^{i\beta}$, where $\alpha,\beta\in[0,2\pi)$. Then $\frac{c^3}{abc}=\frac{abc}{a^3}\frac{abc}{b^3}=\frac1{e^{i\alpha}}\frac1{e^{i\beta}}=e^{i(-(\alpha+\beta))}.$
$$\begin{aligned}x &= e^{i\alpha} + e^{i\beta}+e^{i(-(\alpha+\beta))}\\
&=(\cos\alpha+\cos\beta+\cos(-(\alpha+\beta))) + i(\sin\alpha+\sin\beta+\sin(-(\alpha+\beta)))\end{aligned}$$
Since $$\begin{aligned}&\quad\sin\alpha+\sin\beta+\sin(-(\alpha+\beta))\\
&= 2\sin\frac{\alpha+\beta}2\cos\frac{\alpha-\beta}2-2\sin\frac{\alpha+\beta}2\cos\frac{\alpha+\beta}2\\
&=4\sin\frac{\alpha+\beta}2\sin\frac\alpha2\sin\frac\beta2,
\end{aligned}$$
$x$ is real $\iff$ $\sin\frac{\alpha+\beta}2=0$ or $\sin\frac\alpha2=0$ or $\sin\frac\beta2=0$ $\iff$ $\alpha+\beta=0$ or $\alpha=0$ or $\beta=0$.

*

*$\alpha=0$.
$x=1+2\cos\beta\in[-1,3]$.
Note $1+2\cos\beta$ is attainable as we can set $a=1$, $b=e^{i\frac\beta3}$, $c=1/b$.

*$\beta=0$.
$x=1+2\cos\alpha\in[-1,3]$.

*$\alpha+\beta=0$.
$x=1+2\cos\alpha\in[-1,3]$.

Hence, all possible real $x$s are $[-1,3]$.
</s>