ta có :
`(a-b)^2+(b-c)^2+(c-a)^2≥0`
`⇔2a^2+2b^2+2c^2-2ab-2bc-2ac≥0`
`⇔2a^2+2b^2+2c^2≥2ab+2bc+2ac`
`a^2+b^2+c^2≥ab+bc+ac`
áp dụng vào
`⇒a^2+b^2+2c^2≥ab+bc+ac+c^2≥a(b+c)+c(b+c)≥(a+c)(c+b)`
tương tự
`a^2+2b^2+c^2≥(a+b)(c+b)`
`2a^2+b^2+c^2≥(a+c)(a+b) `
`⇒P=≤(ab)/((a+c)(b+c))+(ac)/((a+b)(b+c))+(cb)/((a+c)(b+a))`
`⇒P≤\sqrt(ab)/(a+c)\sqrt(ab)/(b+c)+\sqrt(ac)/(a+b)\sqrt(ac)/(b+c)+\sqrt(cb)/(a+c)\sqrt(cb)/(a+b)`
`⇒P≤1/2 ((ab)/(a+c)+(ab)/(b+c)+(ac)/(a+b)+(ac)/(b+c)+(cb)/(a+c)+(bc)/(b+a))`
`⇒P≤1/2 ((b(a+c))/(a+c)+(a(b+c))/(b+c)+(c(a+b))/(a+b))`
`⇒P≤1/2 (a+b+c)`
`⇒P≤1/2 .6`
`⇒P≤3`
`''=''`xảy ra khi :
`a=b=c=2`
vậy` maxP=3 `khi `a=b=c=2`
