5. Solve the system of equations:
$$
\left\{\begin{array}{l}
\frac{1}{\sqrt{1+2 x^{2}}}+\frac{1}{\sqrt{1+2 y^{2}}}=\frac{2}{\sqrt{1+2 x y}}, \\
\sqrt{x(1-2 x)}+\sqrt{y(1-2 y)}=\frac{2}{9}
\end{array} .\right.
$$
Prompt: First to prove: Equation (1) $\Leftrightarrow x=y$.
Thus, Equation (2) can be transformed into $x(1-2 x)=\frac{1}{81}$.
Solving, we get $x=y=\frac{1}{4} \pm \frac{\sqrt{73}}{36}$.
6. Let non-zero real numbers $a, b, c$ be distinct, and the equations
$$
\begin{array}{l}
a x^{3}+b x+c=0, b x^{3}+c x+a=0, \\
c x^{3}+a x+b=0
\end{array}
$$

have a common root. Prove: At least one of the equations has three real roots.