8. A square $A B C D$ and an equilateral triangle $P Q R$ are inscribed in a circle centred at $O$ in such a way that $A B \| Q R$. The sides $P Q$ and $P R$ of the triangle meet the side $A B$ of the square at $X$ and $Y$ respectively.

The value of $\tan \angle X O Y$ is
(A) $\frac{1}{\sqrt{3}}$;
(B) 1 ;
(C) $\frac{\sqrt{6}-\sqrt{3}}{\sqrt{2}}$;
(D) $\frac{2 \sqrt{2}-2}{\sqrt{3}}$;
(E) $\sqrt{3}$.
8. Answer: (C).

Let the side of the square be 2. Then the radius of the circle is $\sqrt{2}$. Let $\theta=\angle X O Y$. So
$$
\tan (\theta / 2)=\frac{M Y}{M O}=M Y=P M \tan 30^{\circ}=\frac{\sqrt{2}-1}{\sqrt{3}} .
$$

Then
$$
\tan \theta=\frac{2 \tan (\theta / 2)}{1-\tan ^{2}(\theta / 2)}=\frac{\sqrt{6}-\sqrt{3}}{\sqrt{2}} .
$$