Sure, let's write a test case generator for this problem. The generator will create random inputs that conform to the given constraints.

Here is the Python code for the test case generator:

```python
import random

def tcgen():
    # Generate N in the range [2, 100]
    N = random.randint(2, 100)
    
    # Generate x in the range [1, 10^9]
    x = random.randint(1, 10**9)
    
    # Generate list of a_i where each a_i is in the range [1, 10^9]
    A = [random.randint(1, 10**9) for _ in range(N)]
    
    return N, x, A

# Example of using the generator
if __name__ == "__main__":
    N, x, A = tcgen()
    print(N)
    print(x)
    print(" ".join(map(str, A)))
```

This code snippet will generate a random test case each time it runs, ensuring that all constraints are met:

1. `N` is a random integer between 2 and 100.
2. `x` is a random integer between 1 and \(10^9\).
3. `A` is a list of `N` integers, each between 1 and \(10^9\).

You can run the generator to produce various test cases, which can then be used to test your algorithm for correctness and performance.