Question 125
Given an integer $n \gt 0$ and real numbers $x_1 \le x_2 \le \dots \le x_n$, $y_1 \ge y_2 \ge \dots \ge y_n$, satisfying $\sum_{i=1}^n ix_i = \sum_{i=1}^n iy_i$. Prove that for any real number $\alpha$, $\sum_{i=1}^n x_i[i\alpha] \ge \sum_{i=1}^n y_i[i\alpha]$, where $[\beta]$ is defined as the greatest integer less than or equal to $\beta$.