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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

What is the minimum value of $$p(2)$$ if the following 4 conditions are followed?

1. $$p(x)$$ is a polynomial of degree $$17$$.

2. All roots of $$p(x)$$ are real.

3. All coefficients are positive.

4. The coefficient of $$x^{17}$$ is 1 and the product of roots of $$p(x)$$ is 1.

##### Image Credit: Wikimedia Septic Graph

$x_1^2+x_2^2+\cdots +x_n^2 \le \dfrac{1}{x_1^2}+\dfrac{1}{x_2^2}+\cdots +\dfrac{1}{x_n^2}$

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 hold true.

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

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

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

$\large P=(x+y)(z+t)$

Let $$x,y,z,t$$ be positive real numbers satisfying $$x^2+y^2+z^2+t^2=10$$ and $$xyzt=4$$.

If the maximum value of the expression is $$S$$, find $$\lfloor 100 S \rfloor$$.

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 $$a$$.

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