Question 154
Let $x_1, \dots, x_n$ ($n \ge 2$) be real numbers such that
$$
A = \left| \sum_{i=1}^{n} x_i \right| \neq 0
$$
and
$$
B = \max_{1 \le i \lt j \le n} |x_i - x_j| \neq 0.
$$
Prove that for every $n$ vectors $\alpha_1, \dots, \alpha_n$ on the plane, there exists a permutation $(k_1, k_2, \dots, k_n)$ of $(1, 2, \dots, n)$ such that
$$
\left| \sum_{i=1}^{n} x_{k_i} \alpha_i \right| \ge \frac{AB}{2A+B} \max_{1 \le i \le n} |\alpha_i|.
$$
$$
A = \left| \sum_{i=1}^{n} x_i \right| \neq 0
$$
and
$$
B = \max_{1 \le i \lt j \le n} |x_i - x_j| \neq 0.
$$
Prove that for every $n$ vectors $\alpha_1, \dots, \alpha_n$ on the plane, there exists a permutation $(k_1, k_2, \dots, k_n)$ of $(1, 2, \dots, n)$ such that
$$
\left| \sum_{i=1}^{n} x_{k_i} \alpha_i \right| \ge \frac{AB}{2A+B} \max_{1 \le i \le n} |\alpha_i|.
$$