Power Mean Inequalities

Power Mean Inequalities: Level 2 Challenges


ab+bc+cd+da\dfrac{a}{b} + \dfrac{b}{c} + \dfrac{c}{d} + \dfrac{d}{a}

If a,b,c a, b, c and d d are any four positive real numbers, then find the minimum value of the expression above.

Given that a,a, b,b, and cc are positive reals such that a+b+c=6a+b+c=6, find the maximum possible value of abcabc.

Find the minimum value of a2+b2+c2{ a }^{ 2 }+{ b }^{ 2 }+{ c }^{ 2 } for positive numbers a,b,ca,b,c satisfying the constraint a+b+c=6a+b+c=6.

For a,b,c>0a,b,c>0 and a+b+c=6a+b+c=6. Find the minimum value of

(a+1b)2+(b+1c)2+(c+1a)2 \large \left (a+\frac{1}{b} \right )^{2}+ \left (b+\frac{1}{c} \right )^{2}+\left (c+\frac{1}{a} \right )^{2}

Let x,yx,y be positive reals such that x+y=2x+y =2, find the maximum value of x3y3(x3+y3)x^{3}y^{3}(x^{3}+y^{3}).


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