ISL 1982 Problem 42

Let $\mathfrak F$ be the family of all $k$ -element subsets of the set $\{1, 2, \ldots, 2k + 1\}$ . Prove that there exists a bijective function $f :\mathfrak F \to \mathfrak F$ such that for every $A \in \mathfrak F$ , the sets $A$ and $f(A)$ are disjoint.