Problem B. 4708. (April 2015)

B. 4708. \(\displaystyle O\) is the centre of the circumscribed circle of triangle \(\displaystyle ABC\), and \(\displaystyle M\) is the orthocentre. Point \(\displaystyle A\) is reflected in the perpendicular bisector of side \(\displaystyle BC\), \(\displaystyle B\) is reflected in the perpendicular bisector of side \(\displaystyle CA\), and finally \(\displaystyle C\) is reflected in the perpendicular bisector of side \(\displaystyle AB\). The reflections are denoted by \(\displaystyle A_1\), \(\displaystyle B_1\), \(\displaystyle C_1\), respectively. Let \(\displaystyle K\) be the centre of the inscribed circle of triangle \(\displaystyle A_1B_1C_1\). Prove that point \(\displaystyle O\) bisects line segment \(\displaystyle MK\).

Suggested by B. Bíró, Eger

(5 pont)

Deadline expired on May 11, 2015.

Statistics:

38 students sent a solution.
5 points:	Baran Zsuzsanna, Bodolai Előd, Cseh Kristóf, Csépai András, Döbröntei Dávid Bence, Fekete Panna, Imolay András, Katona Dániel, Kerekes Anna, Keresztfalvi Bálint, Kocsis Júlia, Kovács 162 Viktória, Kovács Kitti Fanni, Lajkó Kálmán, Lakatos Ádám, Leitereg Miklós, Molnár-Sáska Zoltán, Nagy Ábel, Nagy-György Pál, Németh 123 Balázs, Olexó Gergely, Polgár Márton, Schrettner Bálint, Schwarcz Tamás, Szebellédi Márton, Török Zsombor Áron, Vághy Mihály, Vankó Miléna, Williams Kada.
4 points:	Czirkos Angéla, Gáspár Attila, Glasznova Maja, Sal Kristóf.
3 points:	1 student.
2 points:	2 students.
1 point:	1 student.
0 point:	1 student.

Problems in Mathematics of KöMaL, April 2015