Question 131
Given positive integers $m, n \ge 2$, first select two different $a_i, a_j$ ($j \gt i$) in the integer set $A = \{a_1, a_2, \dots, a_n\}$ and take the difference $a_j - a_i$. Then arrange the $\binom{n}{2}$ differences in ascending order to form a new sequence, which we call 'derived sequence' and is denoted by $\bar{A}$. The number of elements in $\bar{A}$ that can be divided by $m$ is denoted by $\bar{A}(m)$. Prove that for any $m \ge 2$, the corresponding derived sequences $\bar{A}$ and $\bar{B}$, with regard to $A = \{a_1, a_2, \dots, a_n\}$ and $B = \{1, 2, \dots, n\}$, satisfy the inequality $\bar{A}(m) \ge \bar{B}(m)$.