Power Mean Inequalities

Power Mean Inequalities: Level 5 Challenges


\[\large P=(x+y)(z+t)\]

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 \( \lfloor 100 S \rfloor \).

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

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.

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

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.

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.

  5. The product of roots of \(p(x)\) is -1.

Image Credit: Wikimedia Septic Graph

\[\large 16x^2 +y^2 + \frac 1{x^2} + \frac y{2x} \]

Non-zero real numbers \(x\) and \(y\) are such that the minimum value of expression above can be expressed as \(\sqrt m \), then what is \(m\)?


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