"Spoooooooky" Russian rational expressions 26

Notice that: $y=2x$ and $z=4x$ .

$\Rightarrow \dfrac{x^2}{(x-y)(x-z)} + \dfrac{y^2}{(y-x)(y-z)} + \dfrac{z^2}{(z-x)(z-y)}$

$\implies \dfrac{x^2}{(-x)(-3x)}+\dfrac{4x^2}{x(-2x)}+\dfrac{16x^2}{3x(2x)}$

$\implies \dfrac{x^2}{3x^2}+\dfrac{4x^2}{-2x^2}+\dfrac{16x^2}{6x^2}$

$\implies \dfrac{1}{3}-2+\dfrac{8}{3}=\boxed{1}$

Let $\Xi(x,y,z) = (x-y)(y-z)(z-x) \xi = - x^2(y-z) - y^2(z-x) - z^2(x-y)$ .

RHS is order 3. Hence $\xi = c$ , a constant. $Xi(0,1,2) = 2\xi = 0 - 2 + 4 = 2$ .

$\xi(x,y,z) = 1$ as long as $x,y,z$ are distinct.

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I used $\xi$ because I wasn't sure how to get a curly e... and the question is easier with the values provided as $4x = 2y = z$ . Could throw people off it was $1008, 2015, 4031$ for example.