Question 124
Given an integer $n \ge 3$, prove that the set $X = \{1, 2, 3, \dots, n^2 - n\}$ can be divided into two non-intersecting subsets such that neither of them contains $n$ elements $a_1, a_2, \dots, a_n$ with $a_1 \lt a_2 \lt \dots \lt a_n$ and $a_k \le \frac{a_{k-1} + a_{k+1}}{2}$ for all $k = 2, \dots, n-1$.