<s>[INST] find $\left( \frac{x}{x+y} \right)^{2007} + \left( \frac{y}{x+y} \right)^{2007}$ I found this questions from past year maths competition in my country, I've tried any possible way to find it, but it is just way too hard.

if $x, y$ are non-zero numbers satisfying $x^2 + xy + y^2 = 0$, find the value of $$\left(\frac{x}{x+y} \right)^{2007} + \left(\frac{y}{x+y} \right)^{2007}$$
(A). $2$ (B). $1$ (C). $0$ (D). $-1$ (E). $-2$

expanding it would give us $$ \frac { x^{2007} + y^{2007}} {(x+y)^{2007}}$$
how do I calculate this? Very appreciate for all of those who had helped me
 [/INST] Hint: 
We have:
$$A = \left(\frac{x}{x+y} \right)^{2007} + \left(\frac{y}{x+y} \right)^{2007}$$
$$ = \left( \frac{x}{y} \right)^{1003} \frac{x}{x+y} + \left( \frac{y}{x} \right)^{1003}\frac{y}{x+y} $$
$$ = - \left[ \left( \frac{x}{y} \right)^{1002}+ \left( \frac{y}{x} \right)^{1002} \right]$$
From the condition $x^2 + xy + y^2 =0$, we have
 $$\left( \frac{x}{y} \right)^2 + \left( \frac{x}{y} \right) + 1 =0 $$
or,
$$\left( \frac{x}{y} \right)^3 = 1 $$
</s>