# Task 1. (20 points) 

It is known that the only solution to the equation

$$
\pi / 4=\operatorname{arcctg} 2+\operatorname{arcctg} 5+\operatorname{arcctg} 13+\operatorname{arcctg} 34+\operatorname{arcctg} 89+\operatorname{arcctg}(x / 14)
$$

is a natural number. Find it.
# Solution.

Consider the equation:

$$
\operatorname{arcctg} a-\operatorname{arcctg} b=\operatorname{arcctg} y \text {. }
$$

Let $\alpha=\operatorname{arcctg} a, \beta=\operatorname{arcctg} b$. Now $y=\operatorname{ctg}(\alpha-\beta)=\frac{1+\operatorname{ctg} \alpha \operatorname{ctg} \beta}{\operatorname{ctg} \beta-\operatorname{ctg} \alpha}=\frac{1+a b}{b-a}$

$$
\begin{aligned}
\operatorname{arcctg} x / 14 & =\operatorname{arcctg} 1-\operatorname{arcctg} 2-\operatorname{arcctg} 5-\operatorname{arcctg} 13-\operatorname{arcctg} 34-\operatorname{arcctg} 89 \\
& =\operatorname{arcctg} \frac{2+1}{2-1}-\operatorname{arcctg} 5-\operatorname{arcctg} 13-\operatorname{arcctg} 34-\operatorname{arcctg} 89 \\
& =\operatorname{arcctg} 3-\operatorname{arcctg} 5-\operatorname{arcctg} 13-\operatorname{arcctg} 34-\operatorname{arcctg} 89 \\
& =\operatorname{arcctg} \frac{15+1}{5-3}-\operatorname{arcctg} 13-\operatorname{arcctg} 34-\operatorname{arcctg} 89 \\
& =\operatorname{arcctg} 8-\operatorname{arcctg} 13-\operatorname{arcctg} 34-\operatorname{arcctg} 89 \\
& =\operatorname{arcctg} \frac{104+1}{13-8}-\operatorname{arcctg} 34-\operatorname{arcctg} 89 \\
& =\operatorname{arcctg} 21-\operatorname{arcctg} 34-\operatorname{arcctg} 89 \\
& =\operatorname{arcctg} \frac{814+1}{34-21}-\operatorname{arcctg} 89 \\
& =\operatorname{arcctg} 55-\operatorname{arcctg} 89 \\
& =\operatorname{arcctg} \frac{89 * 55+1}{89-55}=\operatorname{arcctg} 144
\end{aligned}
$$

From this, we have $x=144 * 14=2016$.