GR-HMS 2013 Problem 3

We define the sets $A_1,A_2,...,A_{160}$ such that $\left|A_{i} \right|=i$ for all $i=1,2,...,160$ . With the elements of these sets we create new sets $M_1,M_2,...M_n$ by the following procedure: in the first step we choose some of the sets $A_1,A_2,...,A_{160}$ and we remove from each of them the same number of elements. These elements that we removed are the elements of $M_1$ . In the second step we repeat the same procedure in the sets that came of the implementation of the first step and so we define $M_2$ . We continue similarly until there are no more elements in $A_1,A_2,...,A_{160}$ , thus defining the sets $M_1,M_2,...,M_n$ . Find the minimum value of $n$ .