Let $ABCD$ be a square with the side of $20$ units. Amir divides this square into $400$ unit squares. Reza then picks $4$ of the vertices of these unit squares. These vertices lie inside the square $ABCD$ and define a rectangle with the sides parallel to the sides of the square $ABCD.$ There are exactly $24$ unit squares which have at least one point in common with the sides of this rectangle. Find all possible values for the area of a rectangle with these properties.

[hide="Note"][i]Note:[/i] Vid changed to Amir, and Eva change to Reza![/hide]
1. **Understanding the Problem:**
   We are given a square \(ABCD\) with side length \(20\) units, divided into \(400\) unit squares. Reza picks \(4\) vertices of these unit squares to form a rectangle with sides parallel to \(ABCD\). We need to find the possible areas of this rectangle such that exactly \(24\) unit squares have at least one point in common with the sides of the rectangle.

2. **Counting Unit Squares with Common Points:**
   Let the sides of the rectangle be \(a\) and \(b\) unit squares long. The number of unit squares that have at least one point in common with the sides of the rectangle can be calculated as follows:
   - Each side of length \(a\) contributes \(a\) unit squares.
   - Each side of length \(b\) contributes \(b\) unit squares.
   - Each corner of the rectangle is counted twice, so we subtract \(4\) for the four corners.

   Therefore, the total number of unit squares with at least one point in common with the sides of the rectangle is:
   \[
   2(a + b) + 4 - 4 = 2(a + b)
   \]

3. **Setting Up the Equation:**
   Given that exactly \(24\) unit squares have at least one point in common with the sides of the rectangle, we set up the equation:
   \[
   2(a + b) = 24
   \]
   Simplifying, we get:
   \[
   a + b = 12
   \]

4. **Finding Possible Values for \(a\) and \(b\):**
   We need to find pairs \((a, b)\) such that \(a + b = 12\) and both \(a\) and \(b\) are positive integers. The possible pairs are:
   \[
   (1, 11), (2, 10), (3, 9), (4, 8), (5, 7), (6, 6), (7, 5), (8, 4), (9, 3), (10, 2), (11, 1)
   \]

5. **Calculating the Area of the Rectangle:**
   For each pair \((a, b)\), the area of the rectangle is \(a \times b\). We calculate the areas for each pair:
   \[
   \begin{align*}
   (1, 11) & \rightarrow 1 \times 11 = 11 \\
   (2, 10) & \rightarrow 2 \times 10 = 20 \\
   (3, 9) & \rightarrow 3 \times 9 = 27 \\
   (4, 8) & \rightarrow 4 \times 8 = 32 \\
   (5, 7) & \rightarrow 5 \times 7 = 35 \\
   (6, 6) & \rightarrow 6 \times 6 = 36 \\
   (7, 5) & \rightarrow 7 \times 5 = 35 \\
   (8, 4) & \rightarrow 8 \times 4 = 32 \\
   (9, 3) & \rightarrow 9 \times 3 = 27 \\
   (10, 2) & \rightarrow 10 \times 2 = 20 \\
   (11, 1) & \rightarrow 11 \times 1 = 11 \\
   \end{align*}
   \]

6. **Identifying Unique Areas:**
   The unique possible areas of the rectangle are:
   \[
   \{11, 20, 27, 32, 35, 36\}
   \]

The final answer is \( \boxed{ \{11, 20, 27, 32, 35, 36\} } \).