Mathematical and Physical Journal
for High Schools
Issued by the MATFUND Foundation
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# Problem B. 4649. (September 2014)

B. 4649. Let $\displaystyle e_1,e_2,\dots,e_n$ denote different lines on the plane, and let $\displaystyle f$ be a line that is not parallel to any of them. Consider all lines $\displaystyle f_{\alpha}$ parallel to $\displaystyle f$. Let $\displaystyle S_{\alpha}$ be the centre of mass of the points $\displaystyle f_{\alpha}\cap e_1, f_{\alpha}\cap e_2, \ldots, f_{\alpha}\cap e_n$. Show that the points $\displaystyle S_{\alpha}$ are collinear.

Suggested by B. Csikós, Budapest

(6 pont)

Deadline expired on October 10, 2014.

### Statistics:

 57 students sent a solution. 6 points: Baran Zsuzsanna, Bodolai Előd, Bursics Balázs, Cseh Kristóf, Csépai András, Englert Franciska, Fekete Panna, Gáspár Attila, Hansel Soma, Juhász 326 Dániel, Kerekes Anna, Kovács 246 Benedek, Kovács 972 Márton, Kőrösi Ákos, Lajkó Kálmán, Leitereg Miklós, Nagy Kartal, Nagy-György Pál, Nagy-György Zoltán, Olexó Gergely, Öreg Botond, Porupsánszki István, Schrettner Bálint, Schwarcz Tamás, Szebellédi Márton, Szőke Tamás, Tomcsányi Gergely, Tóth Viktor, Váli Benedek, Varga-Umbrich Eszter, Williams Kada, Zsakó Ágnes. 5 points: Alexy Marcell, Andó Angelika, Geng Máté, Gyulai-Nagy Szuzina, Imolay András, Kátay Tamás, Katona Dániel, Mócsy Miklós, Sal Kristóf, Telek Máté László, Tompa Tamás Lajos. 4 points: 8 students. 3 points: 1 student. 2 points: 1 student. 1 point: 2 students. 0 point: 1 student. Unfair, not evaluated: 1 solution.

Problems in Mathematics of KöMaL, September 2014