Putnam 2011 B5

Let $a_1, a_2, \dots$ be real numbers. Suppose that there is a constant $A$ such that for all $n$, $$\int_{-\infty}^\infty \left( \sum_{i=1}^n \frac{1}{1 + (x-a_i)^2} \right)^2 dx \leq An.$$ Prove there is a constant $B>0$ such that for all $n$, $$\sum_{i,j=1}^n (1 + (a_i - a_j)^2) \geq Bn^3.$$