Mathematical and Physical Journal
for High Schools
Issued by the MATFUND Foundation
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Problem A. 398. (April 2006)

A. 398. Given a circle k in the plane and a line \ell 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 MXY the intersection point of lines XY and \ell. (If X=Y then take the tangent. If the lines are parallel then MXY is the ideal point.) Construct the second intersection of OMXY 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, \ell and the ideal point of \ell 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;

bX+Y=Y+X for all points X, Y;

cX+O=X for all points X;

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

(5 pont)

Deadline expired on May 18, 2006.


9 students sent a solution.
5 points:Jankó Zsuzsanna, Kisfaludi-Bak Sándor, Nagy 224 Csaba, Paulin Roland, Tomon István.
3 points:1 student.
1 point:2 students.
0 point:1 student.

Problems in Mathematics of KöMaL, April 2006