CWMO 2006 Problem 8

Given a positive integer $n\geq 2$ , let $B_{1}$ , $B_{2}$ , ..., $B_{n}$ denote $n$ subsets of a set $X$ such that each $B_{i}$ contains exactly two elements. Find the minimum value of $\left|X\right|$ such that for any such choice of subsets $B_{1}$ , $B_{2}$ , ..., $B_{n}$ , there exists a subset $Y$ of $X$ such that: (1) $\left|Y\right| = n$ ; (2) $\left|Y \cap B_{i}\right|\leq 1$ for every $i\in\left\{1,2,...,n\right\}$ .