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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



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