Algebra

Power Mean Inequalities

Power Mean Inequalities: Level 4 Challenges

         

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

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.

Given that \(x\), \(y\), and \(z\) are positive real numbers satisfying \(xyz(x+y+z)=1\), minimize \((x+y)(y+z)(z+x)\).

Enter your answer to five decimal places.

\(a, b\) and \(c\) are positive real numbers greater than or equal to 1 satisfying

\[ \begin{align} abc & =100,\\ a^{\lg a} b^{\lg b} c^{\lg c} & \geq 10000. \\ \end{align} \]

What is the value of \(a + b + c \)?

Details and assumptions:

  • \(\lg\) refers to log base 10.

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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