KURSCHAK 1994 Problem 3

Consider the sets $A_1,A_2,\dots,A_n$ . Set $A_k$ is composed of $k$ disjoint intervals on the real axis ( $k=1,2,\dots,n$ ). Prove that from the intervals contained by these sets, one can choose $\left\lfloor\frac{n+1}2\right\rfloor$ intervals such that they belong to pairwise different sets $A_k$ , and no two of these intervals have a common point.