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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.

Power Mean Inequalities: Level 4 Challenges

         

Consider all real numbers that satisfy \[x^3+y^3+z^3-3xyz=1.\] What is the minimum value of \(x^2+y^2+z^2\) to 2 decimal places?

\[\large \frac{1}{\sqrt{5x+1}}+\frac{1}{\sqrt{x^2-3x+4}}\geq\frac{x+3}{2(x+1)}\] How many integer \(x\) satisfy the inequality above

\[\large a^3 + b^3 + c^3 + abc \sqrt5 \] Given that \(a,b\) and \(c\) are the sides of a triangle with perimeter 3. If the minimum value of the expression above can be expressed as \(x + \sqrt y \) for positive integers \(x\) and \(y\), find the value of \(x+ y\).

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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