RMM 2011 Problem 5

For every $n \geq 3$, determine all the configurations of $n$ distinct points $X_{1}, X_{2}, \ldots, X_{n}$ in the plane, with the property that for any pair of distinct points $X_{i}, X_{j}$ there exists a permutation $\sigma$ of the integers $\{1, \ldots, n\}$, such that $\mathrm{d}\left(X_{i}, X_{k}\right)=\mathrm{d}\left(X_{j}, X_{\sigma(k)}\right)$ for all $1 \leq k \leq n$. (We write $\mathrm{d}(X, Y)$ to denote the distance between points $X$ and $Y$.) (United Kingdom) LuKe BetTs