Cho `a,b,c > 0` . Tìm Min của : `P = (x + 3z)/(x+y) + (z + 3x)/(y+z) + (4y)/(z+x)`

xin hay nhất cố làm có tâm lắm rồi :v

ý tưởng :

ta có :`3x+2y+3z=(x+y)+(y+z)+2(x+z)` 

và `3x+2y+3z=2(x+y)+(x+3z)=2(y+z)+(z+3x)=(4y+6(x+z))/2`

ta có :

`P=(x+3z)/(x+y)+(z+3x)/(y+z)+(4y)/(x+z)`

`⇔10+P=((x+3z)/(x+y)+2)+((z+3x)/(y+z)+2)+((4y)/(x+z)+6)`

`⇔10+P=(x+3z+2(x+y))/(x+y)+(z+3x+2(y+z))/(y+z)+(4y+6(x+z))/(x+z)`

 `⇔10+P=(3x+2y+3z)/(x+y)+(3x+2y+3z)/(y+z)+(2(3x+2y+3z))/(x+z)`

`⇔10+P=(3x+2y+3z)(1/(x+y)+1/(y+z)+2/(x+z))`

`⇔10+P=((x+y)+(y+z)+2(x+z))(1/(x+y)+1/(y+z)+2/(x+z))`

theo `bu-nhi-a`

`⇔10+P≥(1.1+1.1+√2.√2)^2`

`⇔10+P≥16`

`⇔P≥6`

`”=”`xẩy ra khi :

`x=y=z`