1. $\left(12\right.$ points) Solve the equation $\left(x^{4}-2\right)\left(2^{\operatorname{tg} x}-1\right)+\left(3^{x^{4}}-9\right) \operatorname{tg} x=0$.
Answer: $\pm \sqrt[4]{2} ; \pi n, \quad n \in Z$.

Solution. The functions $y=2^{t}$ and $y=3^{t}$ are increasing, therefore, the expression $3^{x^{4}}-9=3^{x^{4}}-3^{2}$ has the same sign as $x^{4}-2$, and the expression $2^{\operatorname{tg} x}-1=2^{\operatorname{tg} x}-2^{0}$ has the same sign as $\operatorname{tg} x-0=\operatorname{tg} x$. Thus, the terms in the left-hand side of the equation have the same sign, and equality to zero is possible only if one of the factors is zero. We have $\left[\begin{array}{c}x^{4}-2=0, \\ \operatorname{tg} x=0 .\end{array}\right.$ Solving these equations, we get 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.