DE-BUNDESWETTBEWERB 2014 Problem 8

Three non-collinear points $A_1, A_2, A_3$ are given in a plane. For $n = 4, 5, 6, \ldots$ , $A_n$ be the centroid of the triangle $A_{n-3}A_{n-2}A_{n-1}$ .

[list] a) Show that there is exactly one point $S$ , which lies in the interior of the triangle $A_{n-3}A_{n-2}A_{n-1}$ for all $n\ge 4$ . b) Let $T$ be the intersection of the line $A_1A_2$ with $SA_3$ . Determine the two ratios, $A_1T : TA_2$ and $TS : SA_3$ . [/list]