Algebra
# Power Mean Inequalities

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

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

\[\large \dfrac{1}{a^2}+\dfrac{1}{b^2}+\dfrac{1}{c^2}+\dfrac{2(a^2+b^2+c^2)}{3}\] 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

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

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