Question 138
Let $n$ be a positive integer and let $M = \{1, 2, 3, \dots, n^2 + n + 2\}$. We consider the subsets $A_1, A_2, \dots, A_n$ of $M$, such that for each $k \in \{1, 2, \dots, n\}$ the set $A_k$ has $n^2 + k + 1$ elements. Prove that the intersection of the $n$ subsets contains at least two consecutive integers.