<s>[INST] Let $(x, y, z)\in \Bbb R$, prove that $\big(\frac{2x-y}{x-y}\big)^2+\big(\frac{2y-z}{y-z}\big)^2 +\big(\frac{2z-x}{z-x}\big)^2 \geqslant 5$ 
Olympiad Inequation
Let $x$, $y$ and $c$ be distinct real numbers. Prove that:
$$\left(\frac{2x-y}{x-y}\right)^2+\left(\frac{2y-z}{y-z}\right)^2+\left(\frac{2z-x}{z-x}\right)^2 \geqslant 5.$$

This is an assignment I got from a teacher. I've been baffled by the negative sign in the denominator.
Could you please provide insight as to how would one resolve this inequality please? (Do not solve yet, I've yet to turn in this assignment, I just want to get some tips)
 [/INST] Start with
\begin{eqnarray*}
(2y^2z+2z^2x+2x^2y-yz^2-zx^2-xy^2-3xyz)^2 \geq 0
\end{eqnarray*}
This can be expanded out to give
\begin{eqnarray*}
4 \sum_{cyc} y^2 z^4 -4 \sum_{cyc} y^3 z^3 +\sum_{cyc} y^4 z^2 -4 \sum_{cyc} x y z^4 +\\14 \sum_{cyc} xy^2 z^3 -10 \sum_{cyc} x y^3 z^2 -3x^2 y^2 z^2 \geq 0
\end{eqnarray*}
This can be rearranged to 
\begin{eqnarray*}
(2x-y)^2(y-z)^2(z-x)^2+(x-y)^2(2y-z)^2(z-x)^2+(x-y)^2(y-z)^2(2z-x)^2 \geq 5(x-y)^2(y-z)^2(z-x)^2
\end{eqnarray*}
Now divide by $(x-y)^2(y-z)^2(z-x)^2$ and we have 
\begin{eqnarray*}
\frac{(2x-y)^2}{(x-y)^2}+\frac{(2y-z)^2}{(y-z)^2}+\frac{(2z-x)^2}{(z-x)^2} \geq 5.
\end{eqnarray*}
Edit: 
I worked my way backwards from the result, using the "reduce" algebra package. I then guessed a square that would "consume" some of the terms & got lucky !

(2*x-y)^2*(y-z)^2*(z-x)^2+(x-y)^2*(2*y-z)^2*(z-x)^2+(x-y)^2*(y-z)^2*(2*z-x)^2-5*(x-y)^2*(y-z)^2*(z-x)^2;
(2*(y^2*z+z^2*x+x^2*y)-(yz^2+zx^2+x*y^2)-3*xyz)^2;

</s>