In the diagram, $ ABCD$ is a square, with $ U$ and $ V$ interior points of the sides $ AB$ and $ CD$ respectively. Determine all the possible ways of selecting $ U$ and $ V$ so as to maximize the area of the quadrilateral $ PUQV$.

[img]http://i250.photobucket.com/albums/gg265/geometry101/CMO1992Number3.jpg[/img]
1. **Understanding the Problem:**
   We are given a square \(ABCD\) with points \(U\) and \(V\) on sides \(AB\) and \(CD\) respectively. We need to determine the positions of \(U\) and \(V\) that maximize the area of the quadrilateral \(PUQV\).

2. **Initial Considerations:**
   - The area of quadrilateral \(PUQV\) can be maximized by considering the areas of triangles formed by these points.
   - We need to minimize the sum of areas \(AUP + UBQ\) and maximize the sum of areas \(APD + BQC\).

3. **Using Similar Triangles:**
   - Let \(U\) divide \(AB\) in the ratio \(x\) and \(V\) divide \(CD\) in the ratio \(y\).
   - The perpendicular distances from \(P\) and \(Q\) to the sides of the square are crucial in determining the areas of the triangles.

4. **Perpendicular Distances:**
   - Let the perpendicular from \(Q\) to \(BC\) be \(QR\).
   - By similar triangles, \(QR = \frac{UB \cdot VC}{UB + VC}\).
   - Similarly, let the perpendicular from \(P\) to \(AD\) be \(PS\).
   - By similar triangles, \(PS = \frac{AU \cdot DV}{AU + DV}\).

5. **Maximizing the Sum of Perpendiculars:**
   - We need to maximize the expression \(\frac{UB \cdot VC}{UB + VC} + \frac{AU \cdot DV}{AU + DV}\).
   - Let \(x = UB\) and \(y = VC\). Then, the expression becomes \(\frac{xy}{x + y} + \frac{(1 - x)(1 - y)}{2 - (x + y)}\).

6. **Simplifying the Expression:**
   - The expression simplifies to \(\frac{xy}{x + y} + \frac{(2 - (x + y) + xy - 1)}{2 - (x + y)}\).
   - This further simplifies to \(\frac{xy}{x + y} + \frac{xy - 1}{2 - (x + y)}\).

7. **Using AM-GM Inequality:**
   - For a fixed sum \(x + y\), the product \(xy\) is maximized when \(x = y\) by the AM-GM inequality.
   - Therefore, the expression is maximized when \(x = y\).

8. **Conclusion:**
   - The maximum area of quadrilateral \(PUQV\) occurs when \(U\) and \(V\) are positioned such that \(x = y\).
   - This implies that \(U\) and \(V\) should be at the midpoints of \(AB\) and \(CD\) respectively.

The final answer is \( \boxed{ U } \) and \(V\) should be at the midpoints of \(AB\) and \(CD\) respectively.