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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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