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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.

# Level 3

$$x,y,$$ and $$z$$ are real numbers satisfying $$x^2+y^2 =1$$ and $$y+z=1$$.

Let $$M$$ and $$m$$ be the maximum and minimum values of the expression $$x+2y+z$$, respectively.

Find $$M+m$$.

###### Dedicated to Aditya Sharma.

$\large x=\frac{2z^2}{1+z^2}, \quad y=\frac{2x^2}{1+x^2}, \quad z=\frac{2y^2}{1+y^2}$

Excluding the trivial solution $$(0,0,0)$$, find the number of triplets of real numbers $$(x,y,z)$$ that satisfy the system of equations above.

Let $$y_1, y_2, y_3, \ldots, y_{8}$$ be a permutation of the numbers $$1, 2, 3, \ldots, 8$$. What is the minimum value of $$\displaystyle \sum_{i=1}^8 (y_i + i )^2$$?

For $$a,b,c>0$$ and $$a+b+c=6$$. Find the minimum value of

$\large \left (a+\frac{1}{b} \right )^{2}+ \left (b+\frac{1}{c} \right )^{2}+\left (c+\frac{1}{a} \right )^{2}$

The jelly shop sells its products in two different sets: 3 red jelly cubes and 3 green jelly cuboids.

The 3 red cubes are of side length $$a<b<c$$ while the 3 green cuboids are identical with dimensions $$a, b, c$$ as shown above.

Which option would give you more jelly?

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