ISL 1989 Problem 60

A family of sets $A_1, A_2, \ldots ,A_n$ has the following properties:

(i) Each $A_i$ contains 30 elements. (ii) $A_i \cap A_j$ contains exactly one element for all $i, j, 1 \leq i < j \leq n.$

Determine the largest possible $n$ if the intersection of all these sets is empty.