CN-TST 2014 Problem 2

Let $A$ be a finite set of positive numbers , $B=\{\frac{a+b}{c+d} |a,b,c,d \in A \}$ . Show that: $\left | B \right | \ge 2\left | A \right |^2-1$ , where $|X|$ be the number of elements of the finite set $X$ . (High School Affiliated to Nanjing Normal University )