Sure! Here is a test case generator for the given competitive programming problem:

```python
import random

def tcgen():
    # Generate a random N within the given constraints
    N = random.randint(2, 2 * 10**5)
    
    # Generate the array of (A_i, B_i) pairs
    AB = []
    for _ in range(N):
        A = random.randint(1, 10**9)
        B = random.randint(A, 10**9)
        AB.append((A, B))
    
    return N, AB

# Example usage:
N, AB = tcgen()
print(N)
for a, b in AB:
    print(a, b)
```

Explanation of the generator:

1. **N generation**: It generates an integer \(N\) such that \(2 \leq N \leq 2 \times 10^5\).
2. **(A_i, B_i) pairs generation**: It generates \(N\) pairs of integers (A_i, B_i) such that \(1 \leq A_i \leq B_i \leq 10^9\). Each A_i is a random integer, and B_i is a random integer that is at least A_i to ensure \(A_i \leq B_i\).

Here is the test case generator in action:

```python
# Example usage
if __name__ == "__main__":
    N, AB = tcgen()
    print(N)
    for a, b in AB:
        print(a, b)
```

This will print a valid test case for the problem. Each run will generate a different test case due to the randomness.