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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

$2 \cos^{2}\frac{x}{2} \sin^{2}x = x^{2} + \frac{1}{x^{2}}$

where $$0 \leq x \leq \frac{\pi}{2}$$. What is the number of real root(s) of the above equation?

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?

$\large \dfrac{a+bx^4}{x^2}$

Let $$a$$ and $$b$$ be positive constants. The expression above has a least value when $$x^2=k\sqrt{\dfrac{a}{b}}$$, find $$k$$.

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

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

Let $$x,y$$ and $$z$$ be positive reals such that $$x+y+z=6$$, find the minimum value of the expression

$\dfrac{16yz+36xz+64xy}{xyz}.$

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