Let $ABCDE$ be a regular pentagon. The star $ACEBD$ has area 1. $AC$ and $BE$ meet at $P$, while $BD$ and $CE$ meet at $Q$. Find the area of $APQD$.
1. **Identify the given information and the goal:**
   - We are given a regular pentagon \(ABCDE\).
   - The star \(ACEBD\) has an area of 1.
   - We need to find the area of the quadrilateral \(APQD\), where:
     - \(P\) is the intersection of \(AC\) and \(BE\).
     - \(Q\) is the intersection of \(BD\) and \(CE\).

2. **Understand the structure of the star \(ACEBD\):**
   - The star \(ACEBD\) is formed by connecting every second vertex of the pentagon.
   - The area of the star is given as 1.

3. **Introduce points of intersection and areas:**
   - Let \(R\) be the intersection of \(BE\) and \(AD\).
   - Define \(S_1\) as the area of \(\triangle APR\).
   - Define \(S_2\) as the area of \(\triangle AER\).

4. **Express the area of the star in terms of \(S_1\) and \(S_2\):**
   - The star \(ACEBD\) can be decomposed into smaller triangles. Specifically, it can be seen as consisting of:
     - Four triangles like \(\triangle APR\) (each contributing \(S_1\)).
     - Two triangles like \(\triangle AER\) (each contributing \(S_2\)).
   - Therefore, the total area of the star is:
     \[
     \left[ACEBD\right] = 4S_1 + 2S_2
     \]
   - Given that \(\left[ACEBD\right] = 1\), we have:
     \[
     4S_1 + 2S_2 = 1
     \]

5. **Express the area of \(APQD\) in terms of \(S_1\) and \(S_2\):**
   - The quadrilateral \(APQD\) can be decomposed into:
     - Two triangles like \(\triangle APR\) (each contributing \(S_1\)).
     - One triangle like \(\triangle AER\) (contributing \(S_2\)).
   - Therefore, the area of \(APQD\) is:
     \[
     \left[APQD\right] = 2S_1 + S_2
     \]

6. **Solve for the area of \(APQD\):**
   - From the equation \(4S_1 + 2S_2 = 1\), we can solve for \(2S_1 + S_2\):
     \[
     2S_1 + S_2 = \frac{1}{2}
     \]
   - Therefore, the area of \(APQD\) is:
     \[
     \left[APQD\right] = \frac{1}{2}
     \]

The final answer is \(\boxed{\frac{1}{2}}\).