Algebra
# Power Mean Inequalities

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

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

Given that \(a_1,a_2,a_3,a_4 > 0\), find the maximum constant \(N\) for which the following inequality always holds true:

\[\frac{a_1a_2(a_3^2+a_4^2)+a_3a_4(a_1^2+a_2^2)}{a_1a_2a_3a_4} \geq N.\]

\[\large \dfrac{1}{a^2}+\dfrac{1}{b^2}+\dfrac{1}{c^2}+\dfrac{2(a^2+b^2+c^2)}{3}\] Let \(a,b\) and \(c\) be positive reals satisfy \(a+b+c=3\). Find the minimum value of the expression above.

Give your answer to 2 decimal places

Suppose that the two boxes above have the same surface area and that the three dimensions of the cuboid are not all the same. Which one has a larger volume?

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

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.

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