Problem 5. (20 points) Solve the inequality $202 \sqrt{x^{3}-2 x-\frac{2}{x}+\frac{1}{x^{3}}+4} \leq 0$.

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# Solution.

The inequality $202 \sqrt{x^{3}-2 x-\frac{2}{x}+\frac{1}{x^{3}}+4} \leq 0$ has a solution only if $x^{3}-2 x-\frac{2}{x}+\frac{1}{x^{3}}+4=0$.

Let $t=x+\frac{1}{x}$, then $t^{3}-5 t+4=0, t^{3}-t-4 t+4=0,(t-1)\left(t^{2}+t-4\right)=0, t=1$ or $t=\frac{-1 \pm \sqrt{17}}{2}$.

The equations $x+\frac{1}{x}=1$ and $x+\frac{1}{x}=\frac{-1+\sqrt{17}}{2}$ have no roots.

$$
\begin{gathered}
x+\frac{1}{x}=\frac{-1-\sqrt{17}}{2} \\
x^{2}+\frac{1+\sqrt{17}}{2} x+1=0 \\
D=\left(\frac{1+\sqrt{17}}{2}\right)^{2}-4=\frac{2 \sqrt{17}+2}{4}=\left(\frac{\sqrt{2 \sqrt{17}+2}}{2}\right)^{2} \\
x_{1,2}=\frac{-1-\sqrt{17} \pm \sqrt{2 \sqrt{17}+2}}{4}
\end{gathered}
$$

$$
\text { Answer. } \frac{-1-\sqrt{17} \pm \sqrt{2 \sqrt{17}+2}}{4} \text {. }
$$