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MATH / Let x₁,x₂,…,x₆ be non-negative real numbers such that x₁+x₂+x₃+x₄+x₅+x₆=1, and x₁x₃x₅+x₂x₄x₆≥(1)/(540). Find the maximum value of x₁x₂x₃+x₂x₃x₄+x₃x₄x₅+x₄x₅x₆+x₅x₆x₁+x₆x₁x₂.

Problem

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problem

Let x1,x2,…,x6x_1, x_2, \dots , x_6 be non-negative real numbers such that x1+x2+x3+x4+x5+x6=1x_1 +x_2 +x_3 +x_4 +x_5 +x_6 =1, and x1x3x5+x2x4x6≥1540x_1 x_3 x_5 +x_2 x_4 x_6 \ge \frac{1}{540}. Find the maximum value of x1x2x3+x2x3x4+x3x4x5+x4x5x6+x5x6x1+x6x1x2.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.
Plain-text mathematical notation (without MathML)
Let x₁,x₂,…,x₆ be non-negative real numbers such that x₁+x₂+x₃+x₄+x₅+x₆=1, and x₁x₃x₅+x₂x₄x₆≥(1)/(540). Find the maximum value of
x₁x₂x₃+x₂x₃x₄+x₃x₄x₅+x₄x₅x₆+x₅x₆x₁+x₆x₁x₂.
Original LaTeX notation
Let $x_1, x_2, \dots , x_6$ be non-negative real numbers such that $x_1 +x_2 +x_3 +x_4 +x_5 +x_6 =1$, and $x_1 x_3 x_5 +x_2 x_4 x_6 \ge \frac{1}{540}$. Find the maximum value of
\[x_1 x_2 x_3 + x_2 x_3 x_4 +x_3 x_4 x_5 +x_4 x_5 x_6 +x_5 x_6 x_1 +x_6 x_1 x_2.\]

level

Level 5

type

Intermediate Algebra

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Source and history

task_id
b3fed629-e4b6-5d33-8d23-b7a99458b1b8
task_key
intermediate~5falgebra--test--b3fed629-e4b6-5d33-8d23-b7a99458b1b8
task_revision_id
1
config / split
intermediate_algebra / test

Official source

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