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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.

# Power Mean Inequalities: Level 5 Challenges

$\large (a^2-ab+b^2)(b^2-bc+c^2)(c^2-ac+a^2)\leq\frac{m}{n}$ If $$a,b$$ and $$c$$ are non-negative reals satisfying $$a+b+c=2$$ and $$m$$ and $$n$$ are coprime positive integers, find $$m+n$$.

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 $$a$$.

Let $$x$$, $$y$$ and $$z$$ be nonnegative real numbers such that $$x+y+z=5$$. Find the maximum value of $xy^2 + yz^2 + 2xyz.$

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

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 $$a^3 b^3 c^3$$

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

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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