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Six nonnegative numbers [2011.II.9]

Source:

March 31, 2011

AMCAIMEnumber theoryrelatively prime

Problem Statement

Let

x_1,x_2,\dots ,x_6

be nonnegative real numbers such that

x_1+x_2+x_3+x_4+x_5+x_6=1

, and

x_1x_3x_5+x_2x_4x_6 \geq \frac{1}{540}

. Let

p

and

q

be positive relatively prime integers such that

\frac{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

.

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