At a certain school, the ratio of boys to girls is $1:3$.  Suppose that:
$\bullet$ Every boy has most $2013$ distinct girlfriends.
$\bullet$ Every girl has at least $n$ boyfriends.
$\bullet$ Friendship is mutual.

Compute the largest possible value of $n$.

[i]Proposed by Evan Chen[/i]
1. **Define Variables and Ratios:**
   Let \( b \) be the number of boys and \( g \) be the number of girls. Given the ratio of boys to girls is \( 1:3 \), we have:
   \[
   g = 3b
   \]

2. **Constraints on Friendships:**
   - Each boy has at most 2013 distinct girlfriends.
   - Each girl has at least \( n \) boyfriends.
   - Friendship is mutual.

3. **Total Number of Friendships:**
   - The total number of girlfriends among all boys (including multiplicities) is at most \( 2013b \).
   - The total number of boyfriends among all girls (including multiplicities) is at least \( ng \).

4. **Equating the Two Quantities:**
   Since the total number of friendships must be the same from both perspectives (boys and girls), we have:
   \[
   2013b \geq ng
   \]

5. **Substitute \( g = 3b \) into the Inequality:**
   \[
   2013b \geq n \cdot 3b
   \]

6. **Simplify the Inequality:**
   \[
   2013 \geq 3n
   \]
   \[
   n \leq \frac{2013}{3}
   \]
   \[
   n \leq 671
   \]

7. **Conclusion:**
   The largest possible value of \( n \) is 671. This value is achievable by constructing a bipartite graph \( K_{b, g} \) where \( b = 671 \) and \( g = 2013 \), and each boy is friends with every girl.

The final answer is \( \boxed{671} \).