Algebra inequality problem

Suppose: $x_1 + x_2 + x_3 + x_4 + x_5 + x_6 = 1$ , and $x_1x_3x_5 + x_2x_4x_6 \ge \dfrac {1}{540}$ and $\dfrac{p}{q}$ is the maximum possible value of $x_1x_2x_3 + x_2x_3x_4 + x_3x_4x_5 + x_4x_5x_6 + x_5x_6x_1 + x_6x_1x_2$

Find $p+q$

Details and Assumptions

$x_1, x_2, \dots, x_6$ are non-negative real numbers.

$p$ and $q$ are positive relatively prime integers.

• So you want proof of that? – SalmonKiller May 3 '15 at 16:21
• @SalmonKiller Proof of what exactly? – Arian Tashakkor May 3 '15 at 16:25
• The question asks for an evaluation not a proof.Any hint or solution would be highly appreciated – Arian Tashakkor May 3 '15 at 16:25
• That wasn't really clear from the question. Would you mind editing appropriately? – SalmonKiller May 3 '15 at 16:27
• @SalmonKiller Thanks I wrote the question incompletely – Arian Tashakkor May 3 '15 at 16:31