4. Solve the system $\left\{\begin{array}{l}x^{2}+y^{2} \leq 2, \\ x^{4}-8 x^{2} y^{2}+16 y^{4}-20 x^{2}-80 y^{2}+100=0 .\end{array}\right.$
Answer. $\left(-\frac{\sqrt{2}}{\sqrt{5}} ; \frac{2 \sqrt{2}}{\sqrt{5}}\right),\left(\frac{\sqrt{2}}{\sqrt{5}} ; \frac{2 \sqrt{2}}{\sqrt{5}}\right),\left(\frac{\sqrt{2}}{\sqrt{5}} ;-\frac{2 \sqrt{2}}{\sqrt{5}}\right),\left(-\frac{\sqrt{2}}{\sqrt{5}} ;-\frac{2 \sqrt{2}}{\sqrt{5}}\right)$.

Solution. Transform the equation of the system:

$$
\begin{gathered}
x^{4}-8 x^{2} y^{2}+16 y^{4}-20 x^{2}-80 y^{2}+100=0 \Leftrightarrow x^{4}+8 x^{2} y^{2}+16 y^{4}-20 x^{2}-80 y^{2}+100=16 x^{2} y^{2} \Leftrightarrow \\
\Leftrightarrow\left(x^{2}+4 y^{2}\right)^{2}-20\left(x^{2}+4 y^{2}\right)+100=16 x^{2} y^{2} \Leftrightarrow\left(x^{2}+4 y^{2}-10\right)^{2}=(4 x y)^{2} \Leftrightarrow \\
\Leftrightarrow\left[\begin{array} { l } 
{ x ^ { 2 } + 4 y ^ { 2 } - 1 0 = 4 x y , } \\
{ x ^ { 2 } + 4 y ^ { 2 } - 1 0 = - 4 x y }
\end{array} \Leftrightarrow \left[\begin{array} { l } 
{ ( x - 2 y ) ^ { 2 } = 1 0 , } \\
{ ( x + 2 y ) ^ { 2 } = 1 0 }
\end{array} \Leftrightarrow \left[\begin{array}{l}
x-2 y= \pm \sqrt{10}, \\
x+2 y= \pm \sqrt{10}
\end{array}\right.\right.\right.
\end{gathered}
$$

In each of the four cases, express $x$ and substitute into the inequality.

If $x=2 y+\sqrt{10}$, then $y^{2}+(2 y+\sqrt{10})^{2} \leq 2,5 y^{2}+4 y \sqrt{10}+8 \leq 0,(y \sqrt{5}+2 \sqrt{2})^{2} \leq 0, y=-\frac{2 \sqrt{2}}{\sqrt{5}}$. Then $x=\frac{\sqrt{2}}{\sqrt{5}}$.

The other cases are considered similarly. In the end, we get 4 solutions: $\left(-\frac{\sqrt{2}}{\sqrt{5}} ; \frac{2 \sqrt{2}}{\sqrt{5}}\right),\left(\frac{\sqrt{2}}{\sqrt{5}} ; \frac{2 \sqrt{2}}{\sqrt{5}}\right)$, $\left(\frac{\sqrt{2}}{\sqrt{5}} ;-\frac{2 \sqrt{2}}{\sqrt{5}}\right),\left(-\frac{\sqrt{2}}{\sqrt{5}} ;-\frac{2 \sqrt{2}}{\sqrt{5}}\right)$.