1. $\left(12\right.$ points) Solve the equation $\left(x^{3}-3\right)\left(2^{\operatorname{ctg} x}-1\right)+\left(5^{x^{3}}-125\right) \operatorname{ctg} x=0$.
Answer: $\sqrt[3]{3} ; \frac{\pi}{2}+\pi n, \quad n \in Z$.

Solution. The functions $y=2^{t}$ and $y=5^{t}$ are increasing, therefore, the expression $5^{x^{3}}-125=5^{x^{3}}-5^{3}$ has the same sign as $x^{3}-3$, and the expression $2^{\operatorname{ctg} x}-1=2^{\operatorname{ctg} x}-2^{0}$ has the same sign as $\operatorname{ctg} x-0=\operatorname{ctg} x$. Thus, the terms in the left part of the equation have the same sign, and equality to zero is possible only when one of the factors is zero. We get $\left[\begin{array}{c}x^{3}-3=0, \\ \operatorname{ctg} x=0 .\end{array}\right.$ Solving these equations, we obtain the answer.

Evaluation criteria. For a fully justified correct solution - 12 points. If the participant equated each term to zero without justification and obtained the correct answer, give 4 points.