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### Problem Analysis:

We are tasked to solve the problem where two two-digit numbers, \( AB \) and \( CD \), add up to a three-digit number \( 1CE \). Here:
- \( AB = 10A + B \)
- \( CD = 10C + D \)
- \( 1CE = 100 + 10C + E \)
  
The digits \( A, B, C, D, E \) are distinct.

We must determine the value of \( A \), the tens digit of \( AB \).

---

### Steps:

1. **Formulate the equation for the sum:**
   \[
   AB + CD = 1CE
   \]
   Substituting the expanded forms:
   \[
   (10A + B) + (10C + D) = 100 + 10C + E
   \]
   Simplify:
   \[
   10A + B + 10C + D = 100 + 10C + E
   \]
   Cancel \( 10C \) from both sides:
   \[
   10A + B + D = 100 + E
   \]
   Rearrange:
   \[
   10A = 100 + E - B - D
   \]

2. **Key Observations:**
   - Since \( AB \) and \( CD \) are two-digit numbers, \( A \geq 1 \) and \( C \geq 1 \).
   - The result \( 1CE \) must start with \( 1 \), so \( 100 \leq AB + CD < 200 \).

3. **Range for \( A \):**
   From the equation:
   \[
   10A = 100 + E - B - D
   \]
   The right-hand side must be divisible by 10. Since \( E, B, \) and \( D \) are distinct digits, possible values for \( A \) must be checked.

---

### Trial-and-Error with Logical Elimination:

Let us consider \( AB + CD = 1CE \) while ensuring the constraints on distinct digits.

- \( A = 9 \):
  \[
  10(9) + B + D = 100 + E
  \]
  This simplifies to:
  \[
  B + D = E
  \]
  A valid \( B, D, \) and \( E \) configuration is possible.

- \( A = 8 \) or \( 7 \):
  Testing these values does not yield consistent results for \( AB + CD = 1CE \) within the distinct digit constraint.

---

### Conclusion:
The only feasible value for \( A \) is **9**, so the correct answer is:

**(a) 9**

---

Would you like me to explain any specific step in more detail or test alternative scenarios? Let me know!

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### Follow-Up Questions:
1. How do we ensure that all digits \( A, B, C, D, E \) are distinct?
2. Why must \( AB + CD \) fall between 100 and 200?
3. Could there be other solutions with the same \( A \) value?
4. How would the solution change if the digits were not distinct?
5. What is the significance of divisibility by 10 in simplifying the equation?

### Tip:
When solving digit-based problems, always consider the constraints of the number system (e.g., distinct digits, valid ranges) to narrow down possibilities systematically.

If the sum of the two-digit numbers AB and CD is the three-digit number 1CE, where the letters A, B, C, D, E denote distinct digits, then what is the value of A?

Discover how to determine the value of A when the sum of two-digit numbers AB and CD equals the three-digit number 1CE. This problem explores digit-based arithmetic and logical elimination, ideal for Grades 8-10.

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