Sure, I can help you generate test cases for the given problem statement. Below is a Python function to generate random test cases:

```python
import random

def tcgen():
    N = random.randint(1, 100000)
    
    X = set()
    while len(X) < N:
        X.add(random.randint(0, 10**9))
    X = list(X)
    
    robots = [(x, random.randint(1, 10**9)) for x in X]
    
    return N, robots

def format_test_case(N, robots):
    test_case = f"{N}\n"
    for x, l in robots:
        test_case += f"{x} {l}\n"
    return test_case.strip()

# Example usage:
N, robots = tcgen()
print(format_test_case(N, robots))
```

This function generates random test cases within the given constraints:

1. `N` is chosen randomly between 1 and 100,000.
2. Each `X_i` is a unique integer between 0 and 10^9.
3. Each `L_i` is an integer between 1 and 10^9.

The `format_test_case` function formats the generated test case for easy reading and use.

You can call `tcgen` to generate a random test case and then use `format_test_case` to print it in the required format. Here's an example of how to call and print a generated test case:

```python
N, robots = tcgen()
print(format_test_case(N, robots))
```

This will output a randomly generated test case in the proper format used for competitive programming problems.