What is the maximum number of subsets of size $5$, taken from the set $A=\{1,2,3,...,20\}$ such that any $2$ of them share exactly $1$ element.
1. **Claim:** A number cannot be represented in 5 or more subsets.
   - **Proof:** Consider a number, say 1, that appears in more than 5 subsets. Each subset is of size 5, and if 1 appears in more than 5 subsets, then we need at least 21 different numbers to ensure that each subset has exactly one common element with another subset. This is a contradiction since the set \( A \) has only 20 elements. Therefore, a number cannot be represented in 5 or more subsets.

2. **Claim:** The desired number of subsets is less than 17.
   - **Proof:** We use the following reasoning:
     - Each subset has 5 elements.
     - Each element in the set \( A \) can appear in at most 4 other subsets (since any two subsets share exactly one element).
     - Therefore, the total number of subsets is bounded by:
       \[
       \frac{4 \times 20}{5} = 16
       \]
     - This calculation shows that the maximum number of subsets is 16.

3. **Construction:** To achieve the maximum number of subsets, we need to construct 16 subsets such that any two subsets share exactly one element. This construction is non-trivial and requires careful selection of subsets. However, the proof above shows that the maximum number of such subsets is 16.

\(\boxed{Q.E.D}\)