CN-TST 2012 Problem 1

Complex numbers ${x_i},{y_i}$ satisfy $\left| {{x_i}} \right| = \left| {{y_i}} \right| = 1$ for $i=1,2,\ldots ,n$ . Let $x=\frac{1}{n}\sum\limits_{i=1}^n{{x_i}}$ , $y=\frac{1}{n}\sum\limits_{i=1}^n{{y_i}}$ and $z_i=x{y_i}+y{x_i}-{x_i}{y_i}$ . Prove that $\sum\limits_{i=1}^n{\left| {{z_i}}\right|}\leqslant n$ .