SCHWEITZER 1968 Problem 8

Let $n$ and $k$ be given natural numbers, and let $A$ be a set such that $$|A| \leq \frac{n(n+1)}{k+1}.$$ For $i=1,2,...,n+1$ , let $A_i$ be sets of size $n$ such that $$|A_i \cap A_j| \leq k \;(i \not=j)\ ,$$ $$A= \bigcup_{i=1}^{n+1} A_i.$$ Determine the cardinality of $A$ .

K. Corradi