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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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