2011 AIME II Problems/Problem 9

Let $x_1, x_2, ... , x_6$ be nonnegaative real numbers such that $x_1 +x_2 +x_3 +x_4 +x_5 +x_6 =1$ , and $x_1 x_3 x_5 +x_2 x_4 x_6 \le \frac{1}{540}$ . Let p and q be positive relatively prime integers such that \frac{p}{q} is the maximum possible value of $x_1 x_2 x_3 + x_2 x_3 x_4 +x_3 x_4 x_5 +x_4 x_5 x_6 +x_5 x_6 x_1 +x_6 x_1 x_2$ . Find p+q.