<s>[INST] Prove that : $a^n+b^n+c^n=x^n+y^n+z^n$; $\forall n\in \mathbb{N}$ $a;b;c;x;y;z \in \mathbb{R}$ such that : \begin{matrix}
a+b+c=x+y+z & \\ 
a^2+b^2+c^2=x^2+y^2+z^2 & \\ 
a^3+b^3+c^3=x^3+y^3+z^3 & 
\end{matrix}

Prove that : $a^n+b^n+c^n=x^n+y^n+z^n$; $\forall n\in \mathbb{N}$

P/s : I don't have any ideas about this problem..!!
Thanks :)
 [/INST] Hint: 
Let $p_1=a+b+c, \; q_1 = ab+bc + ca,\; r_1 = abc, \; p_2 = x+y+z, \; q_2 = xy+yz+zx, \; r_2 = xyz$.  Then it is easy to show that:
$$p_1 = p_2, \quad q_1 = q_2, \quad r_1 = r_2$$
Further, $a^n+b^n + c^n$ and $x^n + y^n + z^n$ are both symmetric functions and hence can be expressed in terms of $p_1, q_1, r_1$ and $p_2, q_2, r_2$ respectively in exactly the same manner... so they are equal.
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