Question

cho a,b, c thuộc R biết a^2+b^2+c^2=ab+bc+ca. tính A=(a-b)^2015+(b-c)^2016+(c-a)^2017

Answer 1

\(a^2+b^2+c^2=ab+bc+ca\\ \Leftrightarrow a^2+b^2+c^2-ab-bc-ca=0\\ \Leftrightarrow2a^2+2b^2+2c^2-2ab-2bc-2ca=0\\ \Leftrightarrow\left(a^2-2ab+b^2\right)+\left(b^2-2bc+c^2\right)+\left(c^2-2ca+a^2\right)=0\\ \Leftrightarrow\left(a-b\right)^2+\left(b-c\right)^2+\left(c-a\right)^2=0\\ Vì\left\{{}\begin{matrix}\left(a-b\right)^2\ge0\forall a,b\in R\\\left(b-c\right)^2\ge0\forall b,c\in R\\\left(c-a\right)^2\ge0\forall c,a\in R\end{matrix}\right.\\ \Rightarrow\left\{{}\begin{matrix}\left(a-b\right)^2=0\\\left(b-c\right)^2=0\\\left(c-a\right)^2=0\end{matrix}\right.\\ \Rightarrow a=b=c\\ Khiđó:A=0\)