Algebra

Power Mean Inequalities

Power Mean Inequalities: Level 3 Challenges

         

Given that \(a_1,a_2,a_3,a_4 > 0\), find the maximum constant \(N\) for which the following inequality always holds true:

\[\frac{a_1a_2(a_3^2+a_4^2)+a_3a_4(a_1^2+a_2^2)}{a_1a_2a_3a_4} \geq N.\]

Let \(a\) and \(b\) be positive real numbers such that \(a+b=1\). Find the maximum value of \({ a }^{ b }{ b }^{ a }+{ a }^{ a }{ b }^{ b }\).

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

Give your answer to 2 decimal places

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

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

Is this true?

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