I notice that this problem involves squares of binomials and sums of squares, which makes me think of the identity $(x + y)^2 = x^2 + 2xy + y^2$. If I apply this identity to the first equation, I get $x^2 + 2xy + y^2 = 105$. I can subtract the second equation from this one to eliminate the $x^2$ and $y^2$ terms, and get $2xy = 105 - 65 = 40$. Dividing both sides by 2, I find that $xy = 20$. 
If $(x + y)^2 = 105$ and $x^2 + y^2 = 65$, what is the value of $xy$?