Sure, here is a Python function to generate test cases for the given billiard table problem statement:

```python
import random

def generate_test_case():
    # Generate random number of balls
    n = random.randint(2, 11)
    
    # Generate random dimensions for the table
    w = random.randint(4, 1000)
    h = random.randint(4, 1000)
    
    # Generate random radius for the balls
    r = random.randint(1, 100)
    
    # Generate random initial velocity vector for the hit ball
    while True:
        v_x = random.randint(-10000, 10000)
        v_y = random.randint(-10000, 10000)
        if v_x != 0 or v_y != 0:
            break
    
    # Generate random positions for each ball
    positions = []
    for _ in range(n):
        x = random.randint(r + 1, w - r - 1)
        y = random.randint(r + 1, h - r - 1)
        positions.append((x, y))
    
    return n, w, h, r, v_x, v_y, positions

def generate_test_cases(num_cases):
    test_cases = []
    
    for _ in range(num_cases):
        test_case = generate_test_case()
        test_cases.append(test_case)
    
    return test_cases

def format_test_cases(test_cases):
    output = []
    for case in test_cases:
        n, w, h, r, v_x, v_y, positions = case
        output.append(f"{n}")
        output.append(f"{w} {h} {r} {v_x} {v_y}")
        for pos in positions:
            output.append(f"{pos[0]} {pos[1]}")
    
    output.append("0")
    return "\n".join(output)

# Generate a random number of test cases, for example 5
num_cases = 5
test_cases = generate_test_cases(num_cases)
formatted_test_cases = format_test_cases(test_cases)

print(formatted_test_cases)
```

This function will generate a random number of test cases for the billiard table problem. It first generates the random parameters for each test case, ensuring they fall within the specified constraints. Then, it formats them into the appropriate string format for the input.

Here's how it works:

1. `generate_test_case()`: Generates a single test case with random values within the specified constraints.
2. `generate_test_cases(num_cases)`: Generates a list of test cases.
3. `format_test_cases(test_cases)`: Formats the list of test cases into the required input format.

You can control the number of test cases generated by changing the value of `num_cases` in the example at the end. The generated test cases will be printed in the format specified in the problem statement.