Let $ABCD$ be a square and let the points $M$ on $BC$, $N$ on $CD$, $P$ on $DA$, be such that $\angle (AB,AM)=x,\angle (BC,MN)=2x,\angle (CD,NP)=3x$.
1) Show that for any $0\le x\le 22.5$, such a configuration uniquely exists, and that $P$ ranges over the whole segment $DA$;
2) Determine the number of angles $0\le x\le 22.5$ for which$\angle (DA,PB)=4x$.

(Dan Schwarz)
### Part (a): Construction of the Points

1. **Positioning Point \( M \) on \( BC \)**:
   - Let \( M \) be a point on \( BC \) such that \( \angle BAM = x \). Since \( \angle BAM \leq 22.5^\circ \), \( M \) is uniquely determined on \( BC \).

2. **Reflection to Find Point \( E \)**:
   - Reflect \( AB \) over \( AM \) to find point \( E \) on \( BC \). This ensures that \( AE \) is the reflection of \( AB \) in \( AM \).

3. **Positioning Point \( N \) on \( CD \)**:
   - Place \( N \) on \( CD \) such that \( \angle (BC, MN) = 2x \). Since \( MN \) is perpendicular to \( AE \), \( N \) is uniquely determined on \( CD \).

4. **Positioning Point \( P \) on \( DA \)**:
   - Place \( P \) on \( DA \) such that \( \angle (CD, NP) = 3x \). This ensures that \( NP \) is the reflection of \( AM \) in the perpendicular bisector of \( MN \).

Since \( x \) ranges from \( 0 \) to \( 22.5^\circ \), \( P \) will range over the entire segment \( DA \). This completes the construction and shows that such a configuration uniquely exists for any \( 0 \leq x \leq 22.5^\circ \).

### Part (b): Determining the Number of Angles

1. **Angle Condition**:
   - We need to determine the number of angles \( 0 \leq x \leq 22.5^\circ \) for which \( \angle (DA, PB) = 4x \).

2. **Geometric Relationships**:
   - Given the previous construction, we have:
     \[
     \angle (AB, AM) = x
     \]
     \[
     \angle (BC, MN) = 2x
     \]
     \[
     \angle (CD, NP) = 3x
     \]
   - We need to find \( x \) such that:
     \[
     \angle (DA, PB) = 4x
     \]

3. **Verification**:
   - Since \( \angle (DA, PB) = 4x \) must hold, we need to check the range of \( x \):
     \[
     0 \leq 4x \leq 90^\circ
     \]
     \[
     0 \leq x \leq 22.5^\circ
     \]

4. **Conclusion**:
   - The condition \( \angle (DA, PB) = 4x \) is satisfied for any \( x \) in the given range \( 0 \leq x \leq 22.5^\circ \). Therefore, there are infinitely many angles \( x \) within this range that satisfy the condition.

The final answer is \( \boxed{\text{infinitely many angles}} \).