Question 189
Let $X$ be a set of $n$ different positive integers. Denote by $S(I)$ the sum of all elements in a subset $I$ of $X$ and by $X_m$ the set $\{S(I) : I \subset X \text{ and } |I| = m\}$ for a positive integer $m \leq n$. If $|X_m| = m(n-m)+1$ for an integer $m$ such that $n-1 \gt m \gt 1$ then show that the elements of $X$ form an arithmetic progression.
(Bayarmagnai G.)
(Bayarmagnai G.)