1. Solve the equation

$$
\frac{1}{x+\frac{1}{x^{2}+\frac{1}{x^{3}+\frac{1}{x^{4}}}}}=\frac{1}{1+\frac{1}{x^{3}+\frac{1}{x^{2}+\frac{1}{x^{5}}}}}-\frac{1}{x^{2}+\frac{1}{x+\frac{1}{x^{4}+\frac{1}{x^{3}}}}}
$$

in positive real numbers.
IV/1. We calculate in sequence

$$
\begin{aligned}
& \frac{1}{x+\frac{1}{x^{2}+\frac{1}{x^{3}+\frac{1}{x^{4}}}}}=\frac{1}{x+\frac{1}{x^{2}+\frac{x^{4}}{x^{7}+1}}}=\frac{1}{x+\frac{x^{7}+1}{x^{2}+x^{4}+x^{9}}}=\frac{x^{2}+x^{4}+x^{9}}{1+x^{3}+x^{5}+x^{7}+x^{10}}=a \\
& \frac{1}{1+\frac{1}{x^{3}+\frac{1}{x^{2}+\frac{1}{x^{5}}}}}=\frac{1}{1+\frac{1}{x^{3}+\frac{x^{5}}{x^{7}+1}}}=\frac{1}{1+\frac{x^{7}+1}{x^{3}+x^{5}+x^{10}}}=\frac{x^{3}+x^{5}+x^{10}}{1+x^{3}+x^{5}+x^{7}+x^{10}}=a x \text { and } \\
& \frac{1}{x^{2}+\frac{1}{x+\frac{1}{x^{4}+\frac{1}{x^{3}}}}}=\frac{1}{x^{2}+\frac{1}{x+\frac{x^{3}}{x^{7}+1}}}=\frac{1}{x^{2}+\frac{x^{7}+1}{x+x^{3}+x^{8}}}=\frac{x+x^{3}+x^{8}}{1+x^{3}+x^{5}+x^{7}+x^{10}}=\frac{a}{x}
\end{aligned}
$$

The given equation can thus be written in the form $a=x a-\frac{a}{x}$. Since $a>0$ for $x>0$, we can divide the equation by $a$ and multiply by $x$. We get the quadratic equation $x^{2}-x-1=0$, which has the only positive solution $x=\frac{1+\sqrt{5}}{2}$.

Simplification of each continued fraction: 3 points. Simplification and conversion to the equation $x^{2}-x-1=0$: 3 points. Solution $x=\frac{1+\sqrt{5}}{2}$: 1 point.

If the contestant correctly resolves all fractions and writes a polynomial equation of degree 5 or higher but does not solve it: 3 points. If the contestant partially resolves the obtained equation and correctly writes the solution $x=\frac{1+\sqrt{5}}{2}$ but does not prove that it is the only one: 2 points.