Problem 1596

Given that there are 10 birds on the ground and for any 5 of them, there are at least 4 birds on a circle, we need to determine the least possible number of birds on the circle with the most birds.

To solve this, consider the following steps:

1. Initial Assumption: Let $n$ be the number of birds on the circle with the most birds. We need to find the minimum value of $n$.

2. Case Analysis:
- **Case $n = 10$: All 10 birds are on the same circle. This trivially satisfies the condition.
- Case $n = 9$: Suppose 9 birds are on one circle and 1 bird is outside. For any 5 birds chosen, at least 4 must be on the circle. This condition is satisfied because any set of 5 birds will include at least 4 from the circle of 9.
- Case $n < 9$: If fewer than 9 birds are on the circle, then there are at least 2 birds outside the circle. Consider any 5 birds chosen. If 3 or more of these birds are outside the circle, then fewer than 4 birds will be on the circle, violating the given condition.

3. Conclusion**: The minimum number of birds on the circle with the most birds is 9. This ensures that for any 5 birds chosen, at least 4 will be on the circle, satisfying the condition.

Thus, the least possible number of birds on the circle with the most birds is:
$$\boxed{9}$$