Mathematical and Physical Journal
for High Schools
Issued by the MATFUND Foundation
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# Problem A. 581. (February 2013)

A. 581. In the plane, there are given two circles k1 and k2 with different radii, and a point O lying outside the circles. The end-points of the tangents drawn from O to k1 are P and Q, the end-points of the tangents drawn from O to k2 are R and S. The points P, Q, R, S are distinct. Let H be the external homothety center between k1 and k2. Prove that if PR is not a external common tangent to the circles but passes through H then QS also passes through H.

(5 pont)

Deadline expired on March 11, 2013.

### Statistics:

 17 students sent a solution. 5 points: Bodnár Levente, Di Giovanni Márk, Fehér Zsombor, Herczeg József, Janzer Barnabás, Janzer Olivér, Kabos Eszter, Machó Bónis, Medek Ákos, Nagy Róbert, Omer Cerrahoglu, Sárosdi Zsombor, Szabó 789 Barnabás, Szabó 928 Attila, Williams Kada. 4 points: Maga Balázs, Zilahi Tamás.

Problems in Mathematics of KöMaL, February 2013