Asymmetric Optimization 2

Algebra Level 5

Let $x$ , $y$ and $z$ be three positive real numbers. And, $P$ be the maximum value of $\psi$ , which is defined as follows.

$\psi = \bigg(\frac{x+y}{x+y+z}\bigg)^3 \bigg(\frac{y+z}{x+y+z}\bigg)^2 \bigg(\frac{z+x}{x+y+z}\bigg)$

Find $\lfloor 1000 \cdot P \rfloor$ .

Try this similar problem with more constraints.

1 solution

X X
May 27, 2018

Let $a=\dfrac{x+y}{x+y+z},b=\dfrac{y+z}{x+y+z},c=\dfrac{z+x}{x+y+z},a+b+c=2$ .

By AM-GM, $\dfrac26=\dfrac16\left(\dfrac a3+\dfrac a3+\dfrac a3+\dfrac b2+\dfrac b2+c\right)\ge\left(\dfrac{a^3b^2c}{3^32^2}\right)^{\frac16}$

$3^32^2\left(\dfrac26\right)^6=\dfrac4{27}\ge a^3b^2c$

Thanks for posting a solution. Please, also try "Asymmetric Optimization" which is kind of similar to this one.

Atomsky Jahid - 3 years ago

OK,I will.

X X - 3 years ago

Good Solution!! thanks!!

Dong kwan Yoo - 3 years ago