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 have the following equations involving integers \( a, b, c \):

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

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

### Step 1: Express \( c \) in terms of \( a \) and \( b \)

From the first equation, we can express \( c \) as:
\[ c = a + b - 1 \]

**Hint:** Substitute \( c \) in the second equation to simplify it.

### Step 2: Substitute \( c \) into the second equation

Now, substituting \( c \) into the second equation:
\[ a^2 + b^2 - (a + b - 1)^2 = -1 \]

### Step 3: Expand the equation

Expanding \( (a + b - 1)^2 \):
\[ (a + b - 1)^2 = a^2 + b^2 + 1 + 2ab - 2a - 2b \]
So, we have:
\[ a^2 + b^2 - (a^2 + b^2 + 1 + 2ab - 2a - 2b) = -1 \]

### Step 4: Simplify the equation

This simplifies to:
\[ a^2 + b^2 - a^2 - b^2 - 1 - 2ab + 2a + 2b = -1 \]
\[ -2ab + 2a + 2b - 1 = -1 \]
Adding 1 to both sides gives:
\[ -2ab + 2a + 2b = 0 \]

### Step 5: Factor the equation

Factoring out 2:
\[ 2(-ab + a + b) = 0 \]
This leads to:
\[ -ab + a + b = 0 \]
Rearranging gives:
\[ ab = a + b \]

### Step 6: Rearranging the equation

We can rewrite this as:
\[ ab - a - b = 0 \]
Factoring gives:
\[ (a - 1)(b - 1) = 1 \]

### Step 7: Find integer solutions

The integer pairs \((a - 1, b - 1)\) that multiply to 1 are:
1. \( (1, 1) \) leading to \( a = 2, b = 2 \)
2. \( (-1, -1) \) leading to \( a = 0, b = 0 \)

### Step 8: Calculate \( c \) for each pair

1. For \( (a, b) = (2, 2) \):
\[ c = 2 + 2 - 1 = 3 \]
So, \( a^2 + b^2 + c^2 = 2^2 + 2^2 + 3^2 = 4 + 4 + 9 = 17 \).

2. For \( (a, b) = (0, 0) \):
\[ c = 0 + 0 - 1 = -1 \]
So, \( a^2 + b^2 + c^2 = 0^2 + 0^2 + (-1)^2 = 0 + 0 + 1 = 1 \).

### Step 9: Sum the possible values

The possible values of \( a^2 + b^2 + c^2 \) are 17 and 1. Therefore, the sum of all possible values is:
\[ 17 + 1 = 18 \]

### Final Answer
The sum of all possible values of \( a^2 + b^2 + c^2 \) is \( \boxed{18} \).

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