1. Find all values of $x$, for each of which the expressions

$$
\log _{2011}\left(\sqrt{1+\operatorname{ctg}^{2} x}+\operatorname{ctg} x\right) \quad \text { and } \quad \log _{2013}\left(\sqrt{1+\operatorname{ctg}^{2} x}-\operatorname{ctg} x\right)
$$

are equal to each other.
# Problem 1. 

Answer: $\pi n, n \in \mathbb{Z}$ (in variants IV-1 and IV-3) and $\frac{\pi}{2}+\pi n, n \in \mathbb{Z}$ (in variants IV-2 and IV-4).

Solution. The product of the arguments of the logarithms is 1, so one logarithm is non-negative while the other is non-positive. Therefore, the logarithms coincide only in the case when both arguments are equal to 1. Thus, $\operatorname{tg} x=0$ in variants IV-1 and IV-3 and $\operatorname{ctg} x=0$ in variants IV-2 and IV-4.