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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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