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Power Mean Inequalities

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:

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

Hint: Click here for hint.


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