Problem

Source: Romanian National Olympiad 1998 - Grade 10 - Problem 1

Tags: algebra, Integers, combinatorics



Let $n \ge 2$ be an integer and $M= \{1,2,\ldots,n\}.$ For each $k \in \{1,2,\ldots,n-1\}$ we define $$x_k= \frac{1}{n+1} \sum_{\substack{A \subset M \\ |A|=k}} (\min A + \max A).$$Prove that the numbers $x_k$ are integers and not all of them are divisible by $4.$

HIDE: Notations $|A|$ is the cardinal of $A$ $\min A$ is the smallest element in $A$ $\max A$ is the largest element in $A$