Mathematical and Physical Journal
for High Schools
Issued by the MATFUND Foundation
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# Problem B. 4701. (March 2015)

B. 4701. Let $\displaystyle A_1 B_1 C_1 D_1$ be a quadrilateral. For any set of four points $\displaystyle A_n B_n C_n D_n$ already defined for a positive integer $\displaystyle n$, let $\displaystyle A_{n+1}$ be the centroid of the triangle $\displaystyle B_{n}C_{n}D_{n}$. $\displaystyle B_{n+1}$, $\displaystyle C_{n+1}$ and $\displaystyle D_{n+1}$ are defined analogously, with a cyclic permutation of the points. Show that for any starting quadrilateral, the point sequence $\displaystyle A_n$ only has a finite number of points lying outside the unit circle drawn about the centre of mass of the quadrilateral $\displaystyle A_1 B_1 C_1 D_1$.

Suggested by E. Gáspár Merse, Budapest

(4 pont)

Deadline expired on April 10, 2015.

### Statistics:

 51 students sent a solution. 4 points: Andi Gabriel Brojbeanu, Andó Angelika, Barabás Ábel, Baran Zsuzsanna, Csépai András, Csorba Benjámin, Döbröntei Dávid Bence, Fekete Panna, Gál Hanna, Gáspár Attila, Hansel Soma, Hraboczki Attila Márton, Katona Dániel, Kerekes Anna, Kocsis Júlia, Kovács 162 Viktória, Leitereg Miklós, Molnár-Sáska Zoltán, Nagy Dávid Paszkál, Nagy-György Pál, Németh 123 Balázs, Polgár Márton, Schrettner Bálint, Schwarcz Tamás, Szász Dániel Soma, Szebellédi Márton, Tóth Viktor, Vághy Mihály, Vágó Ákos, Vankó Miléna, Varga-Umbrich Eszter, Wei Cong Wu, Williams Kada, Záhorský Ákos, Zsakó Ágnes. 3 points: Alexy Marcell, Cseh Kristóf, Czirkos Angéla, Gyulai-Nagy Szuzina, Kuchár Zsolt, Lajkó Kálmán, Nagy-György Zoltán, Porupsánszki István. 2 points: 4 students. 0 point: 4 students.

Problems in Mathematics of KöMaL, March 2015