SILK-ROAD 2011 Problem 1

Determine the smallest possible value of $| A_{1} \cup A_{2} \cup A_{3} \cup A_{4} \cup A_{5} |$ , where $A_{1}, A_{2}, A_{3}, A_{4}, A_{5}$ sets simultaneously satisfying the following conditions: $(i)$ $| A_{i}\cap A_{j} | = 1$ for all $1\leq i < j\leq 5$ , i.e. any two distinct sets contain exactly one element in common; $(ii)$ $A_{i}\cap A_{j} \cap A_{k}\cap A_{l} =\varnothing$ for all $1\leq i<j<k<l\leq 5$ , i.e. any four different sets contain no common element. Where $| S |$ means the number of elements of $S$ .