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The problem involves a quadratic equation \( ax^2 + bx + c = 0 \) with equal roots, asking how many distinct integral solutions exist for \( a+b+c \) when \( a, b, \) and \( c \) are distinct two-digit positive numbers and perfect squares.

### Solution Steps:

1. **Equal Roots Condition**: For a quadratic equation \( ax^2 + bx + c = 0 \) to have equal roots, its discriminant must be zero. The discriminant is given by \( b^2 - 4ac \). Setting this equal to zero, we get:
   \[
   b^2 - 4ac = 0
   \]

2. **Perfect Squares Requirement**: Since \( a, b, \) and \( c \) need to be distinct two-digit numbers and perfect squares, let's identify the two-digit perfect squares. They are:
   - \( 16, 25, 36, 49, 64, 81 \)

3. **Finding Combinations**:
   - We iterate through all possible values of \( a, b, \) and \( c \) among the listed perfect squares, ensuring they are distinct, and check which combinations satisfy the condition \( b^2 = 4ac \).

4. **Compute \( a+b+c \)**:
   - For each valid combination of \( a, b, c \) that satisfies \( b^2 = 4ac \), we compute \( a+b+c \) and check if it results in a unique sum.

Let's calculate and find out how many distinct sums \( a+b+c \) can be achieved under these conditions.After computing all possible combinations of \( a, b, c \) from the distinct two-digit perfect squares, there are no combinations where \( a, b, \) and \( c \) are distinct, satisfy \( b^2 = 4ac \), and result in a distinct sum \( a+b+c \). Therefore, the answer to how many distinct integral solutions exist for \( a+b+c \) under these conditions is \( \textbf{0} \).

Would you like more details on any part of this solution, or do you have any other questions?

Here are five related questions to expand on the current topic:

1. What happens if we relax the condition of "perfect squares" and consider all two-digit numbers?
2. How would the number of solutions change if the quadratic equation allowed for two distinct roots instead of equal roots?
3. What is the general behavior of quadratic equations with roots that are perfect squares?
4. Can you provide examples of quadratic equations with roots that are prime numbers?
5. How can the sum of coefficients affect the nature of the roots in a quadratic equation?

**Tip:** When investigating quadratic equations, exploring the discriminant (given by \( b^2 - 4ac \)) is a quick way to understand the nature of the roots, such as whether they are real, equal, or complex.

For a quadratic equation ax^2 + bx + c = 0 with equal roots, how many distinct integral solutions will a + b + c have if a, b, and c are distinct two-digit positive numbers and perfect squares?

Explore the mathematical challenge of finding distinct integral solutions for the sum of coefficients in a quadratic equation with equal roots, where each coefficient is a distinct two-digit perfect square. This advanced problem is suitable for high school students studying algebra.

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