Question 135
Let $n$ be a natural number with $n \ge 2$. Prove that, for any complex numbers $a_1, a_2, \dots, a_n$ and $b_1, b_2, \dots, b_n$, the following statements are equivalent:
$$
a) \quad \sum_{k=1}^{n} |z - a_k|^2 \le \sum_{k=1}^{n} |z - b_k|^2, \text{ for any } z \in \mathbb{C};
$$
$$
b) \quad \sum_{k=1}^{n} a_k = \sum_{k=1}^{n} b_k \text{ and } \sum_{k=1}^{n} |a_k|^2 \le \sum_{k=1}^{n} |b_k|^2.
$$
$$
a) \quad \sum_{k=1}^{n} |z - a_k|^2 \le \sum_{k=1}^{n} |z - b_k|^2, \text{ for any } z \in \mathbb{C};
$$
$$
b) \quad \sum_{k=1}^{n} a_k = \sum_{k=1}^{n} b_k \text{ and } \sum_{k=1}^{n} |a_k|^2 \le \sum_{k=1}^{n} |b_k|^2.
$$