Cho `a , b , c > 0` . Chứng minh rằng : `P = (a(b + c))/(a^2 + (b + c)^2) + (b(a + c))/(b^2 + (a + c)^2) + (c(a + b))/(c^2 + (a + b)^2) ≤ 6/5`

`a^2+(b+c)^2=a^2+((b+c)^2)/4 +(3(b+c)^2)/4≥2(a(b+c))/2 +(3(b+c)^2)/4=(4a(b+c))/4+(3(b+c)^2)/4=(b+c)/4 (4a+3(b+c))`

`⇔1/(a^2+(b+c)^2)≤4/((b+c)(4a+3(b+c)))`

`⇔(a(b+c)/(a^2+(b+c)^2)≤(4a)/(4a+3b+3c)≤1/(25) ((9^2a)/(3a+3b+3c))+a/a)=(27a)/(25a+25b+25c)+1/(25)`

CM tương tự 

`⇒(b(a+c)/(b^2+(a+c)^2)≤(27b)/(25a+25b+25c)+1/(25)`

`⇒(c(b+a)/(c^2+(b+a)^2)≤(27c)/(25a+25b+25c)+1/(25)`

`⇒P≤(27a)/(25a+25b+25c)+1/(25)+(27b)/(25a+25b+25c)+1/(25)+(27c)/(25a+25b+25c)+1/(25)`

`⇒P≤(27)/(25)+3/(25)=(30)/(25)=6/5`

`”=”`xẩy ra khi :

`a=b=c`