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# Power Mean Inequalities

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.

If \(a\) and \(b\) are positive numbers, find the maximum value of \(ab(72-3a-4b) \).

\(a, b\) and \(c\) are positive real numbers greater than or equal to 1 satisfying

\[ \begin{align} abc & =100,\\ a^{\lg a} b^{\lg b} c^{\lg c} & \geq 10000. \\ \end{align} \]

What is the value of \(a + b + c \)?

**Details and assumptions**:

- \(\lg\) refers to log base 10.

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