Integers a, b, c satisfy a + b - c = 1 and `a^2 + b^2 - c^2 = -1`. What is the sum of all possible values of `a^(2) + b^(2) + c^(2)` ?

Answer

To solve the problem, we need to find integers \( a, b, c \) that satisfy the equations:

1. \( a + b - c = 1 \) (Equation 1)
2. \( a^2 + b^2 - c^2 = -1 \) (Equation 2)

We want to find the sum of all possible values of \( a^2 + b^2 + c^2 \).

### Step 1: Rearranging Equation 1
From Equation 1, we can express \( c \) in terms of \( a \) and \( b \):
\[ c = a + b - 1 \]

### Step 2: Substituting \( c \) in Equation 2
Now, substitute \( c \) into Equation 2:
\[ a^2 + b^2 - (a + b - 1)^2 = -1 \]

### Step 3: Expanding the Equation
Expanding \( (a + b - 1)^2 \):
\[ (a + b - 1)^2 = a^2 + b^2 + 2ab - 2a - 2b + 1 \]
So, substituting this back into Equation 2 gives:
\[ a^2 + b^2 - (a^2 + b^2 + 2ab - 2a - 2b + 1) = -1 \]

### Step 4: Simplifying the Equation
Simplifying the left-hand side:
\[ a^2 + b^2 - a^2 - b^2 - 2ab + 2a + 2b - 1 = -1 \]
This simplifies to:
\[ -2ab + 2a + 2b - 1 = -1 \]
Adding 1 to both sides:
\[ -2ab + 2a + 2b = 0 \]

### Step 5: Factoring the Equation
Factoring out 2:
\[ 2(-ab + a + b) = 0 \]
Thus, we have:
\[ -ab + a + b = 0 \]
Rearranging gives:
\[ ab = a + b \]

### Step 6: Rearranging Further
We can rearrange this to:
\[ ab - a - b = 0 \]
Factoring gives:
\[ (a - 1)(b - 1) = 1 \]

### Step 7: Finding Integer Solutions
The integer solutions for \( (a - 1)(b - 1) = 1 \) can be:
1. \( a - 1 = 1 \) and \( b - 1 = 1 \) → \( a = 2, b = 2 \)
2. \( a - 1 = -1 \) and \( b - 1 = -1 \) → \( a = 0, b = 0 \)
3. \( a - 1 = 1 \) and \( b - 1 = -1 \) → \( a = 2, b = 0 \)
4. \( a - 1 = -1 \) and \( b - 1 = 1 \) → \( a = 0, b = 2 \)

### Step 8: Finding Corresponding \( c \) Values
Now we calculate \( c \) for each pair \( (a, b) \):
1. For \( (2, 2) \):
\[ c = 2 + 2 - 1 = 3 \quad \Rightarrow \quad a^2 + b^2 + c^2 = 2^2 + 2^2 + 3^2 = 4 + 4 + 9 = 17 \]
2. For \( (0, 0) \):
\[ c = 0 + 0 - 1 = -1 \quad \Rightarrow \quad a^2 + b^2 + c^2 = 0^2 + 0^2 + (-1)^2 = 0 + 0 + 1 = 1 \]
3. For \( (2, 0) \):
\[ c = 2 + 0 - 1 = 1 \quad \Rightarrow \quad a^2 + b^2 + c^2 = 2^2 + 0^2 + 1^2 = 4 + 0 + 1 = 5 \]
4. For \( (0, 2) \):
\[ c = 0 + 2 - 1 = 1 \quad \Rightarrow \quad a^2 + b^2 + c^2 = 0^2 + 2^2 + 1^2 = 0 + 4 + 1 = 5 \]

### Step 9: Summing All Possible Values
The possible values of \( a^2 + b^2 + c^2 \) are \( 17, 1, 5, 5 \). The unique sums are \( 17 \) and \( 1 \) and \( 5 \).

Thus, the sum of all possible values is:
\[ 17 + 1 + 5 = 23 \]

### Final Answer
The sum of all possible values of \( a^2 + b^2 + c^2 \) is \( \boxed{23} \).