## Task A-1.7.

On the island, there are 20 clubs. Each inhabitant of the island is a member of one or two clubs. Each club has at most 25 members, and for every pair of clubs, there is an inhabitant who is a member of both clubs. Determine the smallest and largest possible number of inhabitants on the island.
## First Solution.

The number of pairs of clubs is $\frac{20 \cdot 19}{2}=190$.

For each pair of clubs, there is a resident who is a member of both clubs. For each pair $(i, j)$ such that $1 \leqslant i<j \leqslant 20$, we select and label with $A_{i j}$ one resident who is a member of the $i$-th and $j$-th club.

The residents $A_{i j}$ are all different due to the condition that each resident can be a member of at most two clubs. This shows that there must be at least 190 residents on the island.

This shows that the total number of residents cannot be greater than $190 + 20 \cdot 6 = 310$. If there are exactly 310 residents, and each resident is a member of at most two clubs, the conditions of the problem are satisfied. Therefore, the maximum possible number of residents is exactly 310 (and not less).