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Algebra

# Power Mean Inequalities: Level 3 Challenges

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

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.


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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