\(a^2+b^2+c^2=ab+bc+ca\)
\(\Rightarrow a^2+b^2+c^2-ab-bc-ca=0\)
\(\Rightarrow2a^2+2b^2+2c^2-2ab-2bc-2ca=0\)
\(\Rightarrow\left(a^2-2ab+b^2\right)+\left(b^2-2bc+c^2\right)+\left(c^2-2ab+a^2\right)=0\)
\(\Rightarrow\left(a-b\right)^2+\left(b-c\right)^2+\left(c-a\right)^2=0\)
\(\Rightarrow\left\{{}\begin{matrix}a-b=0\\b-c=0\\c-a=0\end{matrix}\right.\) \(\Rightarrow a=b=c\)
mà \(a^2+b^2+c^2=ab+bc+ca=3\)
\(\Rightarrow3a^2=3\)
\(\Rightarrow a^2=1\)
\(\Rightarrow\left[{}\begin{matrix}a=1\\a=-1\end{matrix}\right.\)
- Với \(a=b=c=1\)
\(P=a^{22}+b^{23}+c^{24}=1^{22}+1^{23}+1^{24}=3\)
- Với \(a=b=c=-1\)
\(P=a^{22}+b^{23}+c^{24}=\left(-1\right)^{22}+\left(-1\right)^{23}+\left(-1\right)^{24}=1\)
