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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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