For a group of children, it holds that in every trio of children from the group, there is a boy named Adam, and in every quartet, there is a girl named Beata.

How many children can be in such a group at most, and what are their names in that case?

(J. Zhouf)

Hint. If you don't know how to start, consider a specific group of children and check if the given properties hold.
Since in every trio there is some boy, there are at most two girls in the entire group. Since in every quartet there is some girl, there are at most three boys in the entire group. Therefore, a group consisting of two girls and three boys is the largest possible suitable group.

Since in every trio there is some Adam, all three boys must be named Adam. Since in every trio there is some Beata, both girls must be named Beata.

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