<s>[INST] If $a+b+c$, $a^2+b^2+c^2$, $a^3+b^3+c^3$ are real, then so are a,b,c 
Let $a,b,c$ be complex numbers with distinct magnitudes such that $a+b+c$, $a^2+b^2+c^2$, $a^3+b^3+c^3$ are real.
Prove that $a,b,c$ are real numbers as well.

I tried to go for contradiction: WLOG assume $a$ is not real. Since $a+b+c\in \mathbb R$, at least one of the numbers $b$ or $c$ is not real. WLOG $b$ is not real. I couldn't get any contradiction from that.
I also attempted something using elementary symmetric polynomials: let $s=a+b+c$, $q=ab+ac+bc$ and $p=abc$
Since $a^2+b^2+c^2=s^2-2q$, $\;\;q$ is real as well.
Since $a^3+b^3+c^3=s^3-3qs+3p$, $\;\;p$ is real.
What then?
Source: an oral exam at a French engineering school.
 [/INST] Hint: As you've shown, $(x-a)(x-b)(x-c)$ has real coefficients. What can be said about the complex roots of a polynomial with real coefficients?
A more geometric proof would be to first show that if any of $a,b,c$ are real, we can replace that value with any real value and it is still true, so you can assume that $|a|<|b|<|c|=1$ and $c$ is not real, by scaling and replacing any real values with smaller values.
But we can prove inductively (using your realization that $p,q$ are real) that $a^n+b^n+c^n$ is always real. Then pick $N$ so that $|a|^n+|b|^n<\epsilon$ for $n>N$. Then the imaginary part of $c^n$ would have to be less than $\epsilon$ for all $n>N$. Then show this is impossible for a particular $\epsilon$ unless $c$ is real.
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