Algebra

Power Mean Inequalities

Power Mean Inequalities: Level 5 Challenges

         

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

Let x,y,z,tx,y,z,t be positive real numbers satisfying x2+y2+z2+t2=10x^2+y^2+z^2+t^2=10 and xyzt=4xyzt=4.

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

k=1nxk2k=1n1xk2\sum_{k=1}^{n} x_k^2 \le \sum_{k=1}^n \dfrac{1}{x_k^2}

Given that x1,x2,,xnx_1,x_2, \ldots,x_n are positive reals whose sum is nn, find the largest integer nn such that the inequality above always holds true.

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

3(b+c)2a+4a+3c3b+12(bc)2a+3c \large \frac{3(b+c)}{2a} + \frac{4a+3c}{3b} + \frac{12(b-c)}{2a+3c}

If a,ba,b and cc are positive real numbers, find the minimum value of the expression above.

Bonus: Find the values of a,ba,b and cc when the expression above is minimized.

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

  1. p(x)p(x) is a polynomial of degree 1717.

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

  3. All coefficients are positive.

  4. The coefficient of x17x^{17} is 1.

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

Image Credit: Wikimedia Septic Graph

16x2+y2+1x2+y2x\large 16x^2 +y^2 + \frac 1{x^2} + \frac y{2x}

Non-zero real numbers xx and yy are such that the minimum value of expression above can be expressed as m\sqrt m , then what is mm?

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