`a)`

` a^2 + b^2 + c^3 + 3 = 2(a+b+c)`

`\to a^2 + b^2 + c^3 + 3 -2a -2b -2c= 0`

`\to (a^2 -2a+1)+(b^2 -2b+1) + (c^2 -2c+1) = 0`

`\to (a-1)^2 + (b-1)^2 + (c-1)^2=0`

Ta có ` (a-1)^2 \ge0;\ (b-1)^2 \ge0;\ (c-1)^2 \ge 0`

` \to (a-1)^2 + (b-1)^2 + (c-1)^2 \ge 0\ ∀ a,b,c`

Dấu ` =` xảy ra khi ` a= b = c = 1`

`\to` Đpcm

`b)`

` (a+b+c)^2 = 3(ab+bc+ac)`

`\to a^2+b^2 +c^2 +2ab +2bc +2ac = 3ab + 3bc +3ac`

`\to a^2 +b^2 +c^2 - ab - bc - ac  = 0`

`\to 2a^2 +2b^2 + 2c^2 - 2ab - 2bc - 2ac = 0`

`\to (a^2 -2ab + b^2)+(b^2 - 2bc + c^2)+(c^2 - 2ac +a^2) = 0`

`\to (a-b)^2 +(b-c)^2 + (a-c)^2 = 0`

`\to` $\begin{cases} a -  b = 0 \\\\ b- c = 0 \\\\ c - a = 0 \end{cases}$

`\to a = b =c`

`\to` Đpcm

`c)`

` (a-b)^2 + (b-c)^2 + (c-a)^2 = (a+b-2c)^2 + (b+c -2a)^2 + (c+a-2b)^2`

`\to [ ( a-b)^2 - (a+b-2c)^2] + [ (b-c)^2 - (b+c -2a)^2] + [ (c-a)^2  - (c+a-2b)^2] = 0`

`\to (a-b+a+b-2c)(a-b-a-b+2c) + ( b-c+b+c-2a)(b-c-b-c+2a) + (c-a+c+a-2b)(c-a-c-a+2b) = 0`

`\to ( 2a -2c)(2c-2b) + ( 2b - 2a)(2a - 2c) + ( 2c - 2b)(2b - 2a) = 0`

`\to 2. [ (a-c)(c-b) + (b-a)(a-c) + (c-b)(b-a)] = 0`

`\to (a-c)(c-b) + (b-a)(a-c) + (c-b)(b-a) = 0`

`\to ac - ab - c^2 + bc + ab - bc - a^2 +ac + bc - ac - b^2 +ab = 0`

`\to (ac-ac+ac) + (ab-ab+ab) + (bc - bc+bc) - a^2 - b^2 - c^2 = 0`

`\to ab + ac + bc - a^2 - b^2 - c^2 = 0`

`\to a^2 + b^2 + c^2 - ab - ac - bc = 0`

`\to 2a^2 +2b^2 + 2c^2 - 2ab - 2bc - 2ac = 0`

`\to (a^2 -2ab + b^2)+(b^2 - 2bc + c^2)+(c^2 - 2ac +a^2) = 0`

`\to (a-b)^2 +(b-c)^2 + (a-c)^2 = 0`

`\to` $\begin{cases} a -  b = 0 \\\\ b- c = 0 \\\\ c - a = 0 \end{cases}$

`\to a = b =c`

`\to` Đpcm

$\begin{array}{l} a)\\ {a^2} + {b^2} + {c^2} + 3 = 2\left( {a + b + c} \right)\\  \Leftrightarrow \left( {{a^2} - 2a + 1} \right) + \left( {{b^2} - 2b + 1} \right) + \left( {{c^2} - 2c + 1} \right) = 0\\  \Leftrightarrow {\left( {a - 1} \right)^2} + {\left( {b - 1} \right)^2} + {\left( {c - 1} \right)^2} = 0\\  \Leftrightarrow \left\{ \begin{array}{l} a - 1 = 0\\ b - 1 = 0\\ c - 1 = 0 \end{array} \right. \Leftrightarrow a = b = c = 1\\ b)\\ {\left( {a + b + c} \right)^2} = 3\left( {ab + bc + ca} \right)\\  \Leftrightarrow {a^2} + {b^2} + {c^2} + 2ab + 2bc + 2ca = 3ab + 3bc + 3ca\\  \Leftrightarrow {a^2} - ab + {b^2} - bc + {c^2} - ca = 0\\  \Leftrightarrow 2{a^2} - 2ab + 2{b^2} - 2bc + 2{c^2} - 2ca = 0\\  \Leftrightarrow \left( {{a^2} - 2ab + {b^2}} \right) + \left( {{b^2} - 2bc + {c^2}} \right) + \left( {{c^2} - 2ca + {a^2}} \right) = 0\\  \Leftrightarrow {\left( {a - b} \right)^2} + {\left( {b - c} \right)^2} + {\left( {c - a} \right)^2} = 0\\  \Leftrightarrow \left\{ \begin{array}{l} a - b = 0\\ b - c = 0\\ c - a = 0 \end{array} \right. \Leftrightarrow a = b = c \end{array}$  

c)

$\begin{array}{l} VT = {\left( {a - b} \right)^2} + {\left( {b - c} \right)^2} + {\left( {c - a} \right)^2}\\ VT = {\left( {a + b} \right)^2} - 4ab + {\left( {b + c} \right)^2} - 4bc + {\left( {c + a} \right)^2} - 4ac\left( 1 \right)\\ VP = {\left( {a + b - 2c} \right)^2} + {\left( {b + c - 2a} \right)^2} + {\left( {c + a - 2b} \right)^2}\\  = {\left[ {\left( {a + b} \right) - 2c} \right]^2} + {\left[ {\left( {b + c} \right) - 2a} \right]^2} + {\left[ {\left( {c + a} \right) - 2b} \right]^2}\\  = {\left( {a + b} \right)^2} - 4c\left( {a + b} \right) + 4{c^2} + {\left( {b + c} \right)^2} - 4a\left( {b + c} \right) + 4{a^2} + {\left( {c + a} \right)^2} - 4b\left( {c + a} \right) + 4{b^2}\left( 2 \right)\\ \left( 1 \right),\left( 2 \right) \Rightarrow  - 4c\left( {a + b} \right) + 4{c^2} - 4a\left( {b + c} \right) + 4{a^2} - 4b\left( {c + a} \right) + 4{b^2} =  - 4ab - 4bc - 4ca\\  \Rightarrow ab - \left( {a + b} \right)c + {c^2} + bc - \left( {b + c} \right)a + {a^2} + ac - \left( {a + c} \right)c + {b^2} = 0\\  \Rightarrow ab - ac - bc + {c^2} + bc - ab - ac + {a^2} + ac - ab - bc + {b^2} = 0\\  \Rightarrow {a^2} + {b^2} + {c^2} - ab - bc - ca = 0\\  \Rightarrow {\left( {a - b} \right)^2} + {\left( {b - c} \right)^2} + {\left( {c - a} \right)^2} = 0\\  \Rightarrow \left\{ \begin{array}{l} a - b = 0\\ b - c = 0\\ c - a = 0 \end{array} \right. \Leftrightarrow a = b = c \end{array}$