<s>[INST] System of two symmetric equations I'm trying to solve the following system of equations over positive reals, but I've got stuck.
$$
\left\{ 
\begin{array}{c}
ab + bc + ca = 12 \\ 
a + b + c + 2 = abc \\ 
\end{array}
\right. 
$$
What I've tried: I've used Vieta's formulas to convert this system of equations to $x^3 + (a_0 + 2)x^2 + 12x + a_0$ by assuming that $a, b, c$ are roots of this polynomial ($a_0$ is just some parameter) and noticed that $abc \le 8$ due to $AM \ge GM$ for ${ab, bc, ca}$. 
 [/INST] Let $a+b+c=3u$, $ab+ac+bc=3v^2$ and $abc=w^3$.
Hence, the our conditions give $v^2=4$ and $w^3=3u+2$.
In another hand,
$$(a-b)^2(a-c)^2(b-c)^2=27(2u^2v^4-4v^6-4u^3w^3+6uv^2w^3-w^6)\geq0,$$
which gives
$$(u-2)^2(2u+5)(6u+13)\leq0$$ or $u=2$, which gives $w^3=8$ and $a$, $b$ and $c$ are roots of the equation
$$x^3-6x^2+12x-8=0$$ or
$$(x-2)^3=0,$$
which gives the answer $a=b=c=2$.
</s>