Chứng minh rằng a=b=c nếu có điều kiện sau: (a+b+c)^2 = 3(a^2

* Hướng dẫn giải

$\begin{array}{l}{\left(a+b+c\right)}^2=3\left(a^2+b^2+c^2\right)\\ a^2+b^2+c^2+2ab+2bc+2ca=3a^2+3b^2+3c^2\\ 2a^2+2b^2+2c^2=2ab+2bc+2ca\\ \left(a^2-2ab+b^2\right)+\left(b^2-2bc+c^2\right)+\left(c^2-2ca+a^2\right)=0\\ {\left(a-b\right)}^2+{\left(b-c\right)}^2+{\left(c-a\right)}^2=0\\ \Rightarrow \left\{\begin{array}{l}{\left(a-b\right)}^2=0\\ {\left(b-c\right)}^2=0\\ {\left(c-a\right)}^2=0\end{array}\right.\Rightarrow a=b=c\end{array}$