Có: \(\left\{{}\begin{matrix}x^2+2y+1=0\\y^2+2z+1=0\\z^2+2x+1=0\end{matrix}\right.\)
\(\Rightarrow x^2+2y+1+y^2+2z+1+z^2+2x+1=0\)
\(\Rightarrow\left(x^2+2x+1\right)+\left(y^2+2y+1\right)+\left(z^2+2z+1\right)=0\)
\(\Rightarrow\left(x+1\right)^2+\left(y+1\right)^2+\left(z+1\right)^2=0^{\left(1\right)}\)
Lại có:
\(\left\{{}\begin{matrix}\left(x+1\right)^2\ge0\forall x\\\left(y+1\right)^2\ge0\forall y\\\left(z+1\right)^2\ge0\forall z\end{matrix}\right.\Rightarrow\left(x+1\right)^2+\left(y+1\right)^2+\left(z+1\right)^2\ge0\forall x;y;z^{\left(2\right)}\)
Từ (1) và (2) \(\Rightarrow\left\{{}\begin{matrix}x+1=0\\y+1=0\\z+1=0\end{matrix}\right.\Rightarrow x=y=z=-1\)
Thay \(x=y=z=-1\) vào \(A\), ta được:
\(A=\left(-1\right)^{2000}+\left(-1\right)^{2000}+\left(-1\right)^{2000}\)
\(=1+1+1=3\)
\(\text{#}\mathit{Toru}\)
