In the square $P Q R S$, point $B$ lies on side $R S$, and point $A$ lies on side $S P$. Segments $Q B$ and $R A$ intersect at point $T$, and the cosine of angle $B T R$ is $-0.2$. Find the side of the square, given that $R A=10$, and $Q B=a$.
## Answer

$\frac{2 a}{\sqrt{100+a^{2}-8 a \sqrt{6}}}$.

In rhombus $ABCD$, a circle is drawn through points $B, C, D$ with center at point $O_{1}$, and a circle is drawn through points $A, B, C$ with center at point $O_{2}$. It is known that the ratio of the length of segment $O_{1} O_{2}$ to the length of segment $B O_{2}$ is 3. Find the measure of angle $ABO_{2}$.

## Hint

Form a trigonometric equation in terms of the desired angle.

## Answer

$\arcsin \frac{\sqrt{17}-3}{4}$.