5. Let non-zero complex numbers $x, y$ satisfy $x^{2}+x y+y^{2}=0$, then the value of the algebraic expression $\left(\frac{x}{x+y}\right)^{1990}+\left(\frac{y}{x+y}\right)^{1990}$ is
A. $2^{-1989}$
B. -1
C. 1
D. None of the above answers is correct
Solve:
$\left(\frac{x}{x+y}\right)^{2}=\frac{x^{2}}{x^{2}+2 x y+y^{2}}=\frac{x}{y}$, and $x^{2}+x y+y^{2}=0 \Rightarrow \frac{x}{y}=\omega$ or $\omega^{2}$.
Thus, $\left(\frac{x}{x+y}\right)^{1990}+\left(\frac{y}{x+y}\right)^{1990}=\omega^{995}+\omega^{1990}=\omega^{2}+\omega=-1$. Therefore, the answer is $B$.