Find all integers $n \geq 2$ for which there exists a sequence of $2n$ pairwise distinct points $(P_1, \dots, P_n, Q_1, \dots, Q_n)$ in the plane satisfying the following four conditions: [list=i] [*]no three of the $2n$ points are collinear; 
[*] $P_iP_{i+1} \ge 1$ for all $i = 1, 2, \dots ,n$, where $P_{n+1}=P_1$;
[*] $Q_iQ_{i+1} \ge 1$ for all $i = 1, 2, \dots, n$, where $Q_{n+1} = Q_1$; and
[*] $P_iQ_j \le 1$ for all $i = 1, 2, \dots, n$ and $j = 1, 2, \dots, n$.[/list]

[i]Ray Li[/i]
To solve the problem, we need to find all integers \( n \geq 2 \) for which there exists a sequence of \( 2n \) pairwise distinct points \( (P_1, \dots, P_n, Q_1, \dots, Q_n) \) in the plane satisfying the given conditions. We will analyze the cases for even and odd \( n \) separately.

### Case 1: Even \( n \)

1. **Construction for Even \( n \)**:
   - Consider a square \( ABCD \) with vertices \( A(-0.4, 0.4) \), \( B(0.4, 0.4) \), \( C(0.4, -0.4) \), and \( D(-0.4, -0.4) \).
   - Construct a small circle of radius \( 0.0001 \) around each of these points, labeling them \( \omega_A \), \( \omega_B \), \( \omega_C \), and \( \omega_D \) respectively.
   - Let \( P_i \) be randomly chosen in \( \omega_A \) if \( i \) is odd and in \( \omega_C \) if \( i \) is even.
   - Let \( Q_i \) be randomly chosen in \( \omega_B \) if \( i \) is odd and in \( \omega_D \) if \( i \) is even.

2. **Verification of Conditions**:
   - **No three points are collinear**: The probability that any three points are collinear is zero due to the random selection within the small circles.
   - **Distance \( P_iP_{i+1} \geq 1 \)**: The distance between \( P_i \) and \( P_j \) where \( i \) and \( j \) have different parity is at least \( 0.8\sqrt{2} - 0.0002 > 1 \).
   - **Distance \( Q_iQ_{i+1} \geq 1 \)**: Similarly, the distance between \( Q_i \) and \( Q_j \) where \( i \) and \( j \) have different parity is at least \( 0.8\sqrt{2} - 0.0002 > 1 \).
   - **Distance \( P_iQ_j \leq 1 \)**: The distance between \( P_i \) and \( Q_j \) is at most \( 0.8 + 0.0002 < 1 \).

Thus, the construction works for even \( n \).

### Case 2: Odd \( n \)

1. **Analysis for Odd \( n \)**:
   - Consider the intersection of the two circles centered at \( Q_1 \) and \( Q_2 \) with radius 1.
   - Each \( P_i \) must lie within this intersection.
   - Split the intersection into two sections along the line \( Q_1Q_2 \), calling them \( X \) and \( Y \). These sections include the boundary of the circles and the line \( Q_1Q_2 \) but not the endpoints.

2. **Contradiction**:
   - Suppose \( P_i \in X \). Then \( P_{i+1} \in Y \) by geometric constraints.
   - Continuing this pattern, since \( n \) is odd, we see that \( P_i \) must be in both \( X \) and \( Y \), which is the open segment \( (Q_1Q_2) \).
   - This implies \( |P_1P_2| < 1 \), which contradicts the condition \( P_iP_{i+1} \geq 1 \).

Thus, the construction does not work for odd \( n \).

The final answer is \( \boxed{ n } \) is even.