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This chain of inequalities forms the foundation for many other classical inequalities. See how the four common "means" - arithmetic, geometric, harmonic, and quadratic - relate to each other.

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Suppose \( xyz = 8 \) with \(x, y, z > 0. \) What is the minimum value of \( x + y + z \)?

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What is the largest \( n \) for which

\[ x^3 + y^3 + z^3 \geq nxyz \]

holds for all \( x, y, z > 0 \)?

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What is the largest \( n \) for which

\[ (x + y)(y + z)(x + z) \geq nxyz \]

holds for all positive \( x, y, z \)?

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Suppose \( 2x + 3y + z = 18. \) Find the maximum value of \( xyz \) for \( x, y, z > 0. \)

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