Two distinct 2-digit prime numbers $p,q$ can be written one after the other in 2 different ways to form two 4-digit numbers. For example, 11 and 13 yield 1113 and 1311. If the two 4-digit numbers formed are both divisible by the average value of $p$ and $q$, find all possible pairs $\{p,q\}$.
We are given that two distinct 2-digit prime numbers \( p \) and \( q \) can be written one after the other in 2 different ways to form two 4-digit numbers. For example, 11 and 13 yield 1113 and 1311. If the two 4-digit numbers formed are both divisible by the average value of \( p \) and \( q \), we need to find all possible pairs \( \{p, q\} \).

1. Let the two 4-digit numbers be \( 100p + q \) and \( 100q + p \).
2. The average value of \( p \) and \( q \) is \( \frac{p+q}{2} \).
3. We are given that \( \frac{p+q}{2} \) divides both \( 100p + q \) and \( 100q + p \).

This implies:
\[
\frac{p+q}{2} \mid 100p + q \quad \text{and} \quad \frac{p+q}{2} \mid 100q + p
\]

4. Multiplying both sides by 2, we get:
\[
p+q \mid 200p + 2q \quad \text{and} \quad p+q \mid 200q + 2p
\]

5. Simplifying, we have:
\[
p+q \mid 200p + 2q \quad \Rightarrow \quad p+q \mid 200p + 2q - 2p - 2q = 198p
\]
\[
p+q \mid 200q + 2p \quad \Rightarrow \quad p+q \mid 200q + 2p - 2q - 2p = 198q
\]

6. Since \( p \) and \( q \) are distinct primes, \( \gcd(p+q, p) = \gcd(p+q, q) = 1 \). Therefore, \( p+q \) must divide 198.

7. The divisors of 198 are: \( 1, 2, 3, 6, 9, 11, 18, 22, 33, 66, 99, 198 \).

8. We need to check which of these divisors can be written as the sum of two distinct 2-digit prime numbers.

- \( p+q = 198 \): The maximum sum of two distinct 2-digit primes is \( 97 + 89 = 186 \), so this case is not possible.
- \( p+q = 99 \): Since \( p+q \) is odd, one of them must be even. However, the only even prime is 2, which is not a 2-digit number, so this case is not possible.
- \( p+q = 66 \): We check the pairs of primes:
  - \( 13 + 53 = 66 \)
  - \( 19 + 47 = 66 \)
  - \( 23 + 43 = 66 \)
  - \( 29 + 37 = 66 \)
- \( p+q = 33 \): Since \( p+q \) is odd, one of them must be even. However, the only even prime is 2, which is not a 2-digit number, so this case is not possible.
- \( p+q \leq 22 \): The minimum sum of two distinct 2-digit primes is \( 11 + 13 = 24 \), so these cases are not possible.

Hence, the only possible pairs \( \{p, q\} \) are:
\[
\boxed{\{13, 53\}, \{19, 47\}, \{23, 43\}, \{29, 37\}}
\]