**A. 398.** Given a circle *k* in the plane and a line lying outside *k*. Moreover, a point *O* is given on the circle. Define the binary operator + on the points of *k* as follows. For arbitrary points *X* and *Y*, denote by *M*_{XY} the intersection point of lines *XY* and . (If *X*=*Y* then take the tangent. If the lines are parallel then *M*_{XY} is the ideal point.) Construct the second intersection of *OM*_{XY} and *k*. (If the line is the tangent at *O* then also the second intersection is *O*.) Denote this point by *X*+*Y* (see the *figure*).

Show that operation + can be extended to the union of *k*, and the ideal point of such that the points with this operation form an Abelian group with the unity *O*; i.e. the operation has the following properties:

*a*) (*X*+*Y*)+*Z*=*X*+(*Y*+*Z*) for all points *X*, *Y*, *Z*;

*b*) *X*+*Y*=*Y*+*X* for all points *X*, *Y*;

*c*) *X*+*O*=*X* for all points *X*;

*d*) For an arbitrary point *X* there exists a point *Y* such that *X*+*Y*=*O*.

(5 points)

**Deadline expired on 18 May 2006.**