<s>[INST] System of equations involving complex numbers I'm getting confused to figure this work out. The only thing that came into my head was using AM-GM inequality, but i just get stuck. Here's the problem: 
Let $a,b,c,$ be the complex numbers such that $abc=1$. and
\begin{cases}
a^{20}+b^{20} + c^{20} &= \frac{1}{a^{20}} + \frac{1}{b^{20}} + \frac{1}{c^{20}} \\
a^{17}+b^{17}+c^{17} &= \frac{1}{a^{17}} + \frac{1}{b^{17}} + \frac{1}{c^{17}}\\
a^{2017} + b^{2017}+c^{2017} &= \frac{1}{a^{2017}} + \frac{1}{b^{2017}} + \frac{1}{c^{2017}}\\
\end{cases}
show that $1 \in \{a,b,c\}$.
Please help me to solve this question, any thought would be helpful.
 [/INST] Hint. If $abc=1$ then, for any integer $n$, the equation
$$a^{n}+b^{n} + c^{n}= \frac{1}{a^{n}} + \frac{1}{b^{n}} + \frac{1}{c^{n}}$$
is equivalent to
$$(1-a^n)(1-b^n)(1-c^n)=0.$$
Moreover note that $\gcd(20,17)=1$, $\gcd(2017,17)=1$, and $\gcd(2017,20)=1$.
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