Let $2000 < N < 2100$ be an integer. Suppose the last day of year $N$ is a Tuesday while the first day of year $N+2$ is a Friday. The fourth Sunday of year $N+3$ is the $m$th day of January. What is $m$?

[i]Based on a proposal by Neelabh Deka[/i]
1. **Determine the first day of year \( N+1 \):**
   - Given that the last day of year \( N \) is a Tuesday, the first day of year \( N+1 \) will be the next day, which is a Wednesday.

2. **Identify if year \( N+1 \) is a leap year:**
   - Since the first day of year \( N+2 \) is a Friday, we need to check if \( N+1 \) is a leap year. 
   - If \( N+1 \) is a leap year, it has 366 days, and if it is a common year, it has 365 days.
   - The first day of year \( N+2 \) being a Friday means that the first day of year \( N+1 \) plus 365 days (if common year) or 366 days (if leap year) should result in a Friday.

3. **Calculate the day of the week for the first day of year \( N+2 \):**
   - If \( N+1 \) is a common year:
     \[
     \text{Wednesday} + 365 \text{ days} = \text{Wednesday} + 52 \text{ weeks} + 1 \text{ day} = \text{Thursday}
     \]
   - If \( N+1 \) is a leap year:
     \[
     \text{Wednesday} + 366 \text{ days} = \text{Wednesday} + 52 \text{ weeks} + 2 \text{ days} = \text{Friday}
     \]
   - Since the first day of year \( N+2 \) is a Friday, \( N+1 \) must be a leap year.

4. **Determine the first day of year \( N+3 \):**
   - Since \( N+1 \) is a leap year, the first day of year \( N+2 \) is a Friday.
   - Year \( N+2 \) is a common year, so it has 365 days.
   - The first day of year \( N+3 \) will be:
     \[
     \text{Friday} + 365 \text{ days} = \text{Friday} + 52 \text{ weeks} + 1 \text{ day} = \text{Saturday}
     \]

5. **Calculate the fourth Sunday of January in year \( N+3 \):**
   - The first day of January in year \( N+3 \) is a Saturday.
   - The first Sunday will be on January 2nd.
   - The fourth Sunday will be:
     \[
     \text{January 2nd} + 3 \times 7 \text{ days} = \text{January 2nd} + 21 \text{ days} = \text{January 23rd}
     \]

The final answer is \( \boxed{23} \).