An ordered triplet solution (x,y,z) with x,y,z `int` (0,1) and satisfying `x^(2)+y^(2)+z^(2) + 2xyz=1` is

Options

A. `(cospi/6, cos (7pi)/18, cos(4pi)/9)`
B. `(cos (2pi)/5, cos(pi)/3, cos pi/10)`
C. `(cos (7pi)/12, cospi/4, cos pi/6)`
D. `(cos pi/12, cos (4pi)/9, cos (17pi)/36)`

Answer

To solve the equation \( x^2 + y^2 + z^2 + 2xyz = 1 \) for ordered triplet solutions \((x, y, z)\) where \(x, y, z \in [0, 1]\), we can follow these steps:

### Step 1: Analyze the Equation
The equation \( x^2 + y^2 + z^2 + 2xyz = 1 \) resembles the equation of a sphere in three dimensions. We need to find integer solutions for \(x\), \(y\), and \(z\) that lie within the interval \([0, 1]\).

### Step 2: Check Boundary Values
Since \(x\), \(y\), and \(z\) can only take values 0 or 1, we can check these combinations:
- \( (0, 0, 0) \)
- \( (0, 0, 1) \)
- \( (0, 1, 0) \)
- \( (0, 1, 1) \)
- \( (1, 0, 0) \)
- \( (1, 0, 1) \)
- \( (1, 1, 0) \)
- \( (1, 1, 1) \)

### Step 3: Substitute Values
We will substitute each combination into the equation to see if it satisfies \( x^2 + y^2 + z^2 + 2xyz = 1 \).

1. For \( (0, 0, 0) \):
\[ 0^2 + 0^2 + 0^2 + 2 \cdot 0 \cdot 0 \cdot 0 = 0 \quad \text{(not a solution)} \]

2. For \( (0, 0, 1) \):
\[ 0^2 + 0^2 + 1^2 + 2 \cdot 0 \cdot 0 \cdot 1 = 1 \quad \text{(solution)} \]

3. For \( (0, 1, 0) \):
\[ 0^2 + 1^2 + 0^2 + 2 \cdot 0 \cdot 1 \cdot 0 = 1 \quad \text{(solution)} \]

4. For \( (0, 1, 1) \):
\[ 0^2 + 1^2 + 1^2 + 2 \cdot 0 \cdot 1 \cdot 1 = 2 \quad \text{(not a solution)} \]

5. For \( (1, 0, 0) \):
\[ 1^2 + 0^2 + 0^2 + 2 \cdot 1 \cdot 0 \cdot 0 = 1 \quad \text{(solution)} \]

6. For \( (1, 0, 1) \):
\[ 1^2 + 0^2 + 1^2 + 2 \cdot 1 \cdot 0 \cdot 1 = 2 \quad \text{(not a solution)} \]

7. For \( (1, 1, 0) \):
\[ 1^2 + 1^2 + 0^2 + 2 \cdot 1 \cdot 1 \cdot 0 = 2 \quad \text{(not a solution)} \]

8. For \( (1, 1, 1) \):
\[ 1^2 + 1^2 + 1^2 + 2 \cdot 1 \cdot 1 \cdot 1 = 5 \quad \text{(not a solution)} \]

### Step 4: Collect Solutions
From our checks, the valid ordered triplet solutions are:
- \( (0, 0, 1) \)
- \( (0, 1, 0) \)
- \( (1, 0, 0) \)

### Final Answer
The ordered triplet solutions \((x, y, z)\) that satisfy the equation \( x^2 + y^2 + z^2 + 2xyz = 1 \) are:
\[ (0, 0, 1), (0, 1, 0), (1, 0, 0) \]