 Determine the least integer $n$ such that for any set of $n$ lines in the 2D plane, there exists either a subset of $1001$ lines that are all parallel, or a subset of $1001$ lines that are pairwise nonparallel.

[i]Proposed by Samuel Wang[/i]

[hide=Solution][i]Solution.[/i] $\boxed{1000001}$
Since being parallel is a transitive property, we note that in order for this to not exist, there must exist at most $1001$ groups of lines, all pairwise intersecting, with each group containing at most $1001$ lines. Thus, $n = 1000^2 + 1 =
\boxed{1000001}$.[/hide]
To determine the least integer \( n \) such that for any set of \( n \) lines in the 2D plane, there exists either a subset of \( 1001 \) lines that are all parallel, or a subset of \( 1001 \) lines that are pairwise nonparallel, we can use the following reasoning:

1. **Understanding the Problem**:
   - We need to find the smallest \( n \) such that any configuration of \( n \) lines in the plane will either have 1001 lines that are all parallel or 1001 lines that are pairwise nonparallel.

2. **Parallel Lines**:
   - If we have a subset of 1001 lines that are all parallel, then they all share the same direction vector.

3. **Pairwise Nonparallel Lines**:
   - If we have a subset of 1001 lines that are pairwise nonparallel, then no two lines in this subset share the same direction vector.

4. **Pigeonhole Principle**:
   - To avoid having 1001 lines that are all parallel, we can have at most 1000 lines in any given direction. This means we can have at most 1000 groups of lines, each group containing lines that are parallel to each other.

5. **Counting the Lines**:
   - If we have 1000 groups of lines, each group containing at most 1000 lines, then the total number of lines is \( 1000 \times 1000 = 1000000 \).

6. **Adding One More Line**:
   - If we add one more line to these 1000000 lines, we will have \( 1000001 \) lines. By the pigeonhole principle, this additional line will either:
     - Belong to one of the existing 1000 groups, making that group contain 1001 parallel lines, or
     - Form a new group, which would mean we have 1001 groups of lines, each containing at least one line, ensuring that there are 1001 pairwise nonparallel lines.

Therefore, the least integer \( n \) such that for any set of \( n \) lines in the 2D plane, there exists either a subset of 1001 lines that are all parallel, or a subset of 1001 lines that are pairwise nonparallel, is \( 1000001 \).

The final answer is \( \boxed{1000001} \).