Question 50
Let $n \ge 3$ be an integer and $M = \{z_1, z_2, \dots, z_n\}$ be a given set of complex numbers with the sum of the elements being different from zero. Starting from the set $M$, we repeatedly apply the following transformation: at one step, we replace each of the $n$ elements of the set from the previous step with the sum of the other elements of the set from the previous step. Is it possible that, after a finite number of steps, we will obtain a set of complex numbers $\{w_1, w_2, \dots, w_n\}$ such that $\sum_{1 \le i \lt j \le n} w_i w_j = \sum_{1 \le i \lt j \le n} z_i z_j$ and $\sum_{1 \le i \le n} w_i \ne \pm \sum_{1 \le i \le n} z_i$?
Dorin Andrica and Sorin Monel Budişan
Dorin Andrica and Sorin Monel Budişan