**B. 3816.** The centre of the inscribed circle of the triangle *ABC* is *O*. The extensions of the line segments *AO*, *BO*, *CO* beyond *O* intersect the circumscribed circle at *A*_{1}, *B*_{1}, *C*_{1}, respectively. Prove that the area of the triangle *A*_{1}*B*_{1}*C*_{1} is

where *R* is the radius of the circumscribed circle, and , , are the angles of the original triangle *ABC*.

(4 points)

**B. 3819.** Show that if *A*_{1}*B*_{1}, *A*_{2}*B*_{2} and *A*_{3}*B*_{3} are three parallel chords of a circle, then the perpendiculars dropped from the point *A*_{1}, *A*_{2} and *A*_{3} onto the line *B*_{2}*B*_{3}, *B*_{3}*B*_{1} and *B*_{1}*B*_{2}, respectively, are concurrent.

(5 points)

**C. 809.** The midpoint of the edge *AE* of the unit cube *ABCDEFGH* is *P*, and the midpoint of the face *BCGF* is *R*.

*a*) Find the area of the intersection of the cube with the plane through the points *P*, *B*, *R*.

*b*) The above plane cuts the cube into two solids. What is the ratio of the volumes of the two parts?

(5 points)