Solve the equation `|{:(x+a,c+b,x+c),(x+b,x+c,x+a),(x+c,x+a,x+b):}|=0`

Answer

To solve the equation \( |{:(x+a,c+b,x+c),(x+b,x+c,x+a),(x+c,x+a,x+b):}|=0 \), we will follow these steps:

### Step 1: Write the Determinant
We start with the determinant:
\[ D = \begin{vmatrix} x + a & x + b & x + c \\ x + b & x + c & x + a \\ x + c & x + a & x + b \end{vmatrix} \]

### Step 2: Apply Column Operations
We can simplify the determinant using column operations. We will perform the operation \( C_1 \to C_1 + C_2 + C_3 \):
\[ D = \begin{vmatrix} 3x + a + b + c & x + b & x + c \\ 3x + b + c + a & x + c & x + a \\ 3x + c + a + b & x + a & x + b \end{vmatrix} \]

### Step 3: Simplify the First Column
Now, let's denote the first column as \( 3x + a + b + c \):
\[ D = \begin{vmatrix} 3x + a + b + c & x + b & x + c \\ 3x + b + c + a & x + c & x + a \\ 3x + c + a + b & x + a & x + b \end{vmatrix} \]

### Step 4: Perform Row Operations
Next, we will perform row operations \( R_2 \to R_2 - R_1 \) and \( R_3 \to R_3 - R_1 \):
\[ D = \begin{vmatrix} 3x + a + b + c & x + b & x + c \\ 0 & (x + c) - (x + b) & (x + a) - (x + c) \\ 0 & (x + a) - (x + b) & (x + b) - (x + c) \end{vmatrix} \]
This simplifies to:
\[ D = \begin{vmatrix} 3x + a + b + c & x + b & x + c \\ 0 & c - b & a - c \\ 0 & a - b & b - c \end{vmatrix} \]

### Step 5: Expand the Determinant
Now, we can expand the determinant along the first column:
\[ D = (3x + a + b + c) \begin{vmatrix} c - b & a - c \\ a - b & b - c \end{vmatrix} \]

### Step 6: Calculate the 2x2 Determinant
Calculating the 2x2 determinant:
\[ \begin{vmatrix} c - b & a - c \\ a - b & b - c \end{vmatrix} = (c - b)(b - c) - (a - c)(a - b) \]
This simplifies to:
\[ (c - b)(b - c) - (a - c)(a - b) = - (c - b)(c - b) + (a - c)(a - b) \]

### Step 7: Set the Determinant to Zero
Setting the determinant \( D = 0 \):
\[ (3x + a + b + c) \left[ (c - b)(b - c) - (a - c)(a - b) \right] = 0 \]

### Step 8: Solve for \( x \)
This gives us two cases:
1. \( 3x + a + b + c = 0 \)
2. The second factor equals zero, which is a separate equation.

From the first case:
\[ 3x = - (a + b + c) \implies x = -\frac{1}{3}(a + b + c) \]

### Final Answer
Thus, the solution to the equation is:
\[ x = -\frac{1}{3}(a + b + c) \]