Mathematical and Physical Journal
for High Schools
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# Problem B. 4754. (December 2015)

B. 4754. Lines $\displaystyle AD$, $\displaystyle BD$ and $\displaystyle CD$ passing through an interior point $\displaystyle D$ of a triangle $\displaystyle ABC$ intersect the opposite sides at $\displaystyle A_{1}$, $\displaystyle B_{1}$ and $\displaystyle C_{1}$, respectively. The midpoints of the segments $\displaystyle A_1B_1$, $\displaystyle B_1C_1$ and $\displaystyle C_1A_1$ are $\displaystyle C_2$, $\displaystyle A_2$ and $\displaystyle B_2$, respectively. Show that the lines $\displaystyle AA_{2}$, $\displaystyle BB_{2}$ and $\displaystyle CC_{2}$ are concurrent.

Proposed by Sz. Miklós, Herceghalom

(5 pont)

Deadline expired on January 11, 2016.

### Statistics:

 50 students sent a solution. 5 points: Andó Angelika, Barabás Ábel, Baran Zsuzsanna, Bodolai Előd, Borbényi Márton, Bukva Balázs, Cseh Kristóf, Döbröntei Dávid Bence, Gáspár Attila, Glasznova Maja, Hansel Soma, Harsányi Benedek, Harsch Leila, Horváth András János, Imolay András, Kerekes Anna, Keresztfalvi Bálint, Klász Viktória, Kocsis Júlia, Kosztolányi Kata, Kovács 162 Viktória, Kovács 246 Benedek, Lajkó Kálmán, Lakatos Ádám, Matolcsi Dávid, Nagy Dávid Paszkál, Németh 123 Balázs, Polgár Márton, Radnai Bálint, Richlik Róbert, Schrettner Bálint, Schrettner Jakab, Szajbély Zsigmond, Szemerédi Levente, Tóth Viktor, Vágó Ákos, Váli Benedek, Varga-Umbrich Eszter. 4 points: Nguyen Viet Hung, Pap Tibor, Wiandt Péter. 3 points: 3 students. 2 points: 2 students. 1 point: 2 students. 0 point: 2 students.

Problems in Mathematics of KöMaL, December 2015