Power Mean Inequalities

Power Mean Inequalities: Level 4 Challenges


Find the minimum value of x6+y654xyx^6 + y^6 -54xy , where xx and yy are real numbers.

Let x,y,zx, y, z be real numbers such that x2+y2+z2=1x^{2}+y^{2}+z^{2}=1. Let the maximum possible value of 6xy+4yz\sqrt {6} xy+4yz be AA.Find A2A^{2}.

This problem is inspired by Joel Tan.

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

Enter your answer to five decimal places.

a,ba, b and cc are positive real numbers greater than or equal to 1 satisfying

abc=100,algablgbclgc10000. \begin{aligned} abc & =100,\\ a^{\lg a} b^{\lg b} c^{\lg c} & \geq 10000. \\ \end{aligned}

What is the value of a+b+ca + b + c ?

Details and assumptions:

  • lg\lg refers to log base 10.

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

6a3+9b3+32c3+14abc 6a^{3} + 9b^{3} + 32c^{3} + \frac{1}{4abc}

Hint: Click here for hint.


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