Twelve people go to a party. First, everybody with no friends at the party leave. Then, at the $i$-th hour, everybody with exactly $i$ friends left at the party leave. After the eleventh hour, what is the maximum number of people left? Note that friendship is mutual.

[i]2017 CCA Math Bonanza Team Round #5[/i]
1. Let's analyze the problem step by step. Initially, there are 12 people at the party.
2. In the first step, everyone with no friends leaves. Since friendship is mutual, if a person has no friends, they leave immediately. However, this step does not affect the maximum number of people left because we are considering the maximum scenario where everyone could potentially have friends.
3. At the $i$-th hour, everyone with exactly $i$ friends leaves. We need to determine the maximum number of people that can remain after the 11th hour.

To understand this, consider the following:
- The maximum number of friends any person can have is 11 (since there are 12 people in total, and friendship is mutual).
- At the 1st hour, everyone with exactly 1 friend leaves.
- At the 2nd hour, everyone with exactly 2 friends leaves.
- This process continues until the 11th hour, where everyone with exactly 11 friends leaves.

4. Since the maximum number of friends a person can have is 11, by the 11th hour, everyone will have left the party. This is because, at each hour, people with a specific number of friends (from 1 to 11) leave, and by the 11th hour, there are no people left who have not been accounted for.

Therefore, after the 11th hour, the maximum number of people left is 0.

The final answer is $\boxed{0}$.