\(a^2+b^2+c^2=ab+bc+ac\)
\(\Leftrightarrow2a^2+2b^2+2c^2-2ab-2bc-2ac=0\)
\(\Leftrightarrow\left(a^2-2ab+b^2\right)+\left(b^2-2bc+c^2\right)+\left(a^2-2ac+c^2\right)=0\)
\(\Leftrightarrow\left(a-b\right)^2+\left(b-c\right)^2+\left(a-c\right)^2=0\)
=>a=b=c
\(\left(a-b\right)^{31}+\left(b-c\right)^{10}+\left(c-a\right)^{2014}\)
\(=\left(a-a\right)^{31}+\left(b-b\right)^{10}+\left(c-c\right)^{2014}\)
\(=0+0+0=0\)