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Algebra

# Power Mean Inequalities: Level 4 Challenges

Let $$x, y, z$$ be real numbers such that $$x^{2}+y^{2}+z^{2}=1$$. Let the maximum possible value of $$\sqrt {6} xy+4yz$$ be $$A$$.Find $$A^{2}$$.

###### This problem is inspired by Joel Tan.

Find the minimum value of $$x^6 + y^6 -54xy$$, where $$x$$ and $$y$$ are real numbers.

Consider all real numbers that satisfy $x^3+y^3+z^3-3xyz=1.$ What is the minimum value of $$x^2+y^2+z^2$$ to 2 decimal places?

$\large { x }^{ 6 }-12{ x }^{ 5 }+a{ x }^{ 4 }+b{ x }^{ 3 }+c{ x }^{ 2 }+dx+64 =0$

Let $$a,b,c,d$$ be all constants.

If all of the roots of the above equation are positive, find $$b + c - a$$.

For positive real numbers $$a, b, c$$, find the minimum integer value possible of the following equation:

$6a^{3} + 9b^{3} + 32c^{3} + \frac{1}{4abc}$