Question

Cho a,b,c là các số thực dương thỏa mãn:\(a^2+b^2+c^2=\left(a-b\right)^2+\left(b-c\right)^2+\left(c-a\right)^2\).

1,Tính a+b+c ,biết rằng ab+bc+ca=9

2,CMR nếu c≥a, c≥b thì c≥a+b

Answer 1

1) \(a^2+b^2+c^2=\left(a-b\right)^2+\left(b-c\right)^2+\left(c-a\right)^2\)

\(\Leftrightarrow a^2+b^2+c^2=a^2-2ab+b^2+b^2-2bc+c^2+c^2-2ca+a^2\)

\(\Leftrightarrow a^2+b^2+c^2-2ab-2bc-2ca=0\)

\(\Leftrightarrow a^2+b^2+c^2+2ab+2bc+2ca-4ab-4bc-4ca=0\)

\(\Leftrightarrow\left(a+b+c\right)^2=4\left(ab+bc+ca\right)=36\)

Mà \(a;b;c\in R^+\Rightarrow a+b+c>0\)

\(\Rightarrow a+b+c=6\)