Giải thích các bước giải:
\(\begin{array}{l}
1,\\
{a^2} + {b^2} + {c^2} + 3 = 2\left( {a + b + c} \right)\\
\Leftrightarrow {a^2} + {b^2} + {c^2} + 3 - 2\left( {a + b + c} \right) = 0\\
\Leftrightarrow \left( {{a^2} - 2a + 1} \right) + \left( {{b^2} - 2b + 1} \right) + \left( {{c^2} - 2c + 1} \right) = 0\\
\Leftrightarrow {\left( {a - 1} \right)^2} + {\left( {b - 1} \right)^2} + {\left( {c - 1} \right)^2} = 0\\
\left. \begin{array}{l}
{\left( {a - 1} \right)^2} \ge 0,\,\,\,\forall a\\
{\left( {b - 1} \right)^2} \ge 0,\,\,\,\forall b\\
{\left( {c - 1} \right)^2} \ge 0,\,\,\,\forall c
\end{array} \right\} \Rightarrow {\left( {a - 1} \right)^2} + {\left( {b - 1} \right)^2} + {\left( {c - 1} \right)^2} \ge 0,\,\,\,\forall a,b,c\\
\Rightarrow \left\{ \begin{array}{l}
{\left( {a - 1} \right)^2} = 0\\
{\left( {b - 1} \right)^2} = 0\\
{\left( {c - 1} \right)^2} = 0
\end{array} \right. \Leftrightarrow \left\{ \begin{array}{l}
a = 1\\
b = 1\\
c = 1
\end{array} \right. \Rightarrow a = b = c = 1\\
2,\\
{\left( {a + b + c} \right)^2} = 3\left( {ab + bc + ca} \right)\\
\Leftrightarrow {a^2} + {b^2} + {c^2} + 2ab + 2bc + 2ca = 3\left( {ab + bc + ca} \right)\\
\Leftrightarrow {a^2} + {b^2} + {c^2} - ab - bc - ca = 0\\
\Leftrightarrow 2{a^2} + 2{b^2} + 2{c^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\\
\left. \begin{array}{l}
{\left( {a - b} \right)^2} \ge 0,\,\,\forall a,b\\
{\left( {b - c} \right)^2} \ge 0,\,\,\,\forall b,c\\
{\left( {c - a} \right)^2} \ge 0,\,\,\,\forall c,a
\end{array} \right\} \Rightarrow {\left( {a - b} \right)^2} + {\left( {b - c} \right)^2} + {\left( {c - a} \right)^2} \ge 0,\,\,\forall a,b,c\\
\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 \left\{ \begin{array}{l}
a = b\\
b = c\\
c = a
\end{array} \right. \Rightarrow a = b = c
\end{array}\)
