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Power Mean Inequalities

This chain of inequalities forms the foundation for many other classical inequalities. See how the four common "means" - arithmetic, geometric, harmonic, and quadratic - relate to each other.

Level 5

         

If the polynomial \(f(x)=4x^4-a.x^3+b.x^2-c.x+5\) where \(a,b,c \in \mathbb{R}\) has \(4\) positive real zeroes(roots) say \(r_1,r_2,r_3, \ and \ r_4\), such that \[\frac{r_1}{2}+\frac{r_2}{4}+\frac{r_3}{5}+\frac{r_4}{8}=1\] Find the value of \(a\).

\[\dfrac{1}{a^4}+\dfrac{1}{b^4}+\dfrac{1}{c^4}=1\]

Given the above equation for positive numbers \(a,b,c\).

Find the minimum value of

\[\dfrac{a^4b^4+a^4c^4+b^4c^4}{a^3b^2c^3}\]

If the minimum value of the above is \(x\), input your answer as \(\lfloor 100x \rfloor\).

This is part of the set Trevor's Ten

Details and Assumptions

  • The answer is not \(300\).

  • It is indeed \(a^3 b^2 c^3\) and not \(a^3 b^3 c^3 \)

\[ \large \frac{3(b+c)}{2a} + \frac{4a+3c}{3b} + \frac{12(b-c)}{2a+3c} \]

If \(a,b\) and \(c\) are positive real numbers, find the minimum value of the expression above.

Bonus: Find the values of \(a,b\) and \(c\) when the expression above is minimized.

\[\large x^5 y + y^5 z + z^5 x \]

Let \(x, y,\) and \(z\) be non-negative reals such that \(x+y+z=1\).

The maximum value of the above expression can be represented as \(\dfrac {a^b}{c^d}\), where \(a\) and \(c\) are not perfect powers, and \(a,b,c,d\) are positive integers. Find the value of \(a+b+c+d\).

\[\] Bonus: Generalize this for the expression \(x^n y + y^n z + z^n x\).

\[\sum_{k=1}^{n} x_k^2 \le \sum_{k=1}^n \dfrac{1}{x_k^2}\]

Given that \(x_1,x_2, \ldots,x_n\) are positive reals whose sum is \(n\), find the largest integer \(n\) such that the inequality above always holds true.

If you think all positive integers \(n\) make the inequality hold true, enter 0 as your answer.

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