aime26-17

Proposed solution

Let's break down the problem step by step:

1. **Understanding the pentagon:**
   - The pentagon \(ABCDE\) has internal angles \(\angle A = \angle E = 90^\circ\) and \(\angle B = \angle D = 45^\circ.\)
   - The pentagon has sides \(AB, BC, CD, DE, AB,\) and points \(B, C,\) and \(D\) lie on the same side of line \(AE.\)

2. **Given values:**
   - The given values are:
     - \(AB = 2026\)
     - \(BC = 14\sqrt2\)
     - \(CD = 2026\)
     - \(DE = 20\)

3. **Finding the area:**
   - The area of pentagon \(ABCDE = 16k\), where \(k\in \mathbb{Z}\).

4. **Finding the possible values:**
   - The number of possible values of \(AB\) is:
   - \(\boxed{2026}\) 

The reasoning behind this problem is that we need to find the number of possible values for \(AB\) in the context of a nonconvex pentagon with internal angles \(\angle A = \angle E = 90^\circ\) and \(\angle B = \angle D = 45^\circ.\) 

The final answer is: 

\(\boxed{2026}\) 

The reasoning behind this problem is that we need to find the number of possible values for \(AB\) in the context of a nonconvex pentagon with internal angles \(\angle A = \angle E = 90^\circ\) and \(\angle B = \angle D = 45^\circ.\) 

The final answer is: 

\(\boxed{2026}\) 

The answer is: 2026