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maximum value is a rational number

Source: Nigerian Mathematics olympiad 2021 round 2 problem 4

February 15, 2021
algebrainequalitiesnumber theoryrelatively prime

Problem Statement

let x1x_1, x2x_2 .... x6x_6 be non-negative reals such that x1+x2+x3+x4+x5+x6=1x_1+x_2+x_3+x_4+x_5+x_6=1 and x1x3x5x_1x_3x_5 + x2x4x6x_2x_4x_6 ≥\geq 1540\frac{1}{540}. Let pp and qq be relatively prime integers such that pq\frac{p}{q} is the maximum value of x1x2x3+x2x3x4+x3x4x5+x4x5x6+x5x6x1+x6x1x2x_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+qp+q
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