3.6 The number of distinct pairs of real numbers $(x, y)$ that satisfy the equations $x=x^{2}+y^{2}, y=2 x y$ is
(A) 0 .
(B) 1 .
(C) 2 .
(D) 3 .
(E) 4 .
(29th American High School Mathematics Examination, 1978)
[Solution] If $y \neq 0$, the second equation gives $x=\frac{1}{2}$, substituting into the first equation yields
$$
y= \pm \frac{1}{2} .
$$

If $y=0$, from the first equation we get $x=0$ or 1.
In summary, the system of equations has solutions $\left(\frac{1}{2}, \pm \frac{1}{2}\right),(0,0),(1,0)$.
Therefore, the answer is $(E)$.