Power Mean Inequalities

Power Mean Inequalities: Level 3 Challenges


Given that a1,a2,a3,a4>0a_1,a_2,a_3,a_4 > 0, find the maximum constant NN for which the following inequality always holds true:

a1a2(a32+a42)+a3a4(a12+a22)a1a2a3a4N.\frac{a_1a_2(a_3^2+a_4^2)+a_3a_4(a_1^2+a_2^2)}{a_1a_2a_3a_4} \geq N.

Let aa and bb be positive real numbers such that a+b=1a+b=1. Find the maximum value of abba+aabb{ a }^{ b }{ b }^{ a }+{ a }^{ a }{ b }^{ b }.

1a2+1b2+1c2+2(a2+b2+c2)3\large \dfrac{1}{a^2}+\dfrac{1}{b^2}+\dfrac{1}{c^2}+\dfrac{2(a^2+b^2+c^2)}{3} Let a,ba,b and cc be positive reals satisfy a+b+c=3a+b+c=3. Find the minimum value of the expression above.

Give your answer to 2 decimal places

For some positive reals satisfying i=124xi=1\displaystyle{ \sum_{i=1}^{24} x_i=1}, determine the maximum possible value of (i=124xi)(i=12411+xi).\displaystyle{ \left (\sum_{i=1}^{24}\sqrt{x_i} \right ) \left (\sum_{i=1}^{24} \frac{1}{\sqrt{1+x_i}} \right )}.

There exist 3 positive numbers such that their sum is 8 and their product is 27.

Is this true?


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