Mathematical and Physical Journal
for High Schools
Issued by the MATFUND Foundation
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Problem A. 655. (November 2015)

A. 655. Two circles, \(\displaystyle k_1\) and \(\displaystyle k_2\) meet at points \(\displaystyle A\) and \(\displaystyle B\). Points \(\displaystyle C\) and \(\displaystyle D\) lie on \(\displaystyle k_1\), while points \(\displaystyle E\) and \(\displaystyle F\) lie on circle \(\displaystyle k_2\) in such a way that \(\displaystyle A\), \(\displaystyle C\), \(\displaystyle E\) are collinear and \(\displaystyle B\), \(\displaystyle D\), \(\displaystyle F\) are collinear, too. Points \(\displaystyle G\) and \(\displaystyle H\) are other two points on lines \(\displaystyle ACE\) and \(\displaystyle BDF\), respectively. The line \(\displaystyle CH\) meets \(\displaystyle FG\) and \(\displaystyle k_1\) the second time at \(\displaystyle I\) and \(\displaystyle J\), respectively. The line \(\displaystyle DG\) meets \(\displaystyle EH\) and \(\displaystyle k_1\) the second time at \(\displaystyle K\) and \(\displaystyle L\), respectively. Circle \(\displaystyle k_2\) meets the lines \(\displaystyle EHK\) and \(\displaystyle FGI\) the second time at \(\displaystyle M\) and \(\displaystyle N\), respectively. The points \(\displaystyle A,B,C,\ldots,N\) are distinct. Show that \(\displaystyle I\), \(\displaystyle J\), \(\displaystyle K\), \(\displaystyle L\), \(\displaystyle M\) and \(\displaystyle N\) are either concyclic or collinear.

(5 pont)

Deadline expired on December 10, 2015.


9 students sent a solution.
5 points:Baran Zsuzsanna, Cseh Kristóf, Gáspár Attila, Lajkó Kálmán, Williams Kada.
4 points:Bukva Balázs.
3 points:2 students.
2 points:1 student.

Problems in Mathematics of KöMaL, November 2015