Chủ đề này có 1 trả lời

[DOTOANNANG]

Với $x,\,y$ dương thì:

 

$$\frac{1}{x}+ \frac{1}{x+ 2\,y}+ \frac{1}{2\,x+ y}\geqq \frac{25}{3\left ( 4\,x+ y \right )}$$

 

Spoiler

$\frac{1}{x}+ \frac{1}{x+ 2\,y}+ \frac{1}{2\,x+ y}-\frac{25}{3\left ( 4\,x+ y \right )}= \frac{2\left ( x- y \right )^{2}\left ( 5\,x+ 3\,y \right )}{3\,x\left ( 2\,x+ y \right )\left ( 4\,x+ y \right )\left ( x+ 2\,y \right )}\geqq 0$

[Arthur Pendragon]

$\frac{1}{3x}+\frac{1}{3x}+\frac{1}{3x}+\frac{1}{x+2y}+\frac{1}{2x+y} \geq \frac{5}{\sqrt[5]{3x.3x.3x.(x+2y)(2x+y)}} \geq \frac{25}{3x+3x+3x+x+2y+2x+y}=\frac{25}{3(4x+y)}$ (???)