HMMT 2011 S3

Let $x$ and $y$ be complex numbers such that $|x|=|y|=1$. (a) [15] Determine the maximum value of $|1+x|+|1+y|-|1+x y|$. (b) [15] Determine the maximum value of $|1+x|+|1+x y|+\left|1+x y^{2}\right|+\ldots+\left|1+x y^{2011}\right|-1006|1+y|$.