<s>[INST] Find $xy+yz+zx$ given systems of three homogenous quadratic equations for $x, y, z$ This is a question from Math Olympiad.

If $\{x,y,z\}\subset\Bbb{R}^+$ and if $$x^2 + xy + y^2 = 3 \\ y^2 + yz + z^2 = 1 \\ x^2 + xz + z^2 = 4$$ find the value of $xy+yz+zx$.

I basically do not know how to approach this question. Please let me know how to approach this question, and if you attach full explanation, I will appreciate it. Thanks.
 [/INST] The given equations $(1),(2),(3)$ can be expressed as:
$$\begin{cases} x^3-y^3=3(x-y) \\ y^3-z^3=y-z \\ z^3-x^3=4(z-x)\end{cases} \stackrel{+}{\Rightarrow} x=3z-2y \ \ (4)$$
Plug $(4)$ in $(1)$ and consider it with $(2)$:
$$\begin{cases} 3z^2-3zy+y^2=1 \\ y^2+yz+z^2=1\end{cases} \Rightarrow 2z^2-4zy=0 \Rightarrow z=2y \ \ (5)$$
Put $(5)$ to $(4)$ to find $x=4y$.
Now you can put $z=2y$ and $x=4y$ to $(1)$ to find $y=\frac{1}{\sqrt{7}}$. Hence:
$$xy+yz+zx=\frac{4}{7}+\frac{2}{7}+\frac{8}{7}=2.$$
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