KöMaL 2012 A576

Find all positive integers $n$, nonzero reals $c_1,c_2,\ldots,c_n$ and real $t$ for which there exists a finite, nonempty set $\{A_1,A_2,\ldots,A_n\}$ of points in the plane $S$ and a nonconstant function $f\colon S\to\mathbb{R}$ such that $\sum_{i=1}^n c_if\big(\varphi(A_i)\big)=t$ for every similarity transformation $\varphi$ of the plane $S$.