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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.

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

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