Sure! Here's a test case generator for the given problem statement:

```python
import random

def tcgen():
    # Generate a random number of contests between 1 and 100
    N = random.randint(1, 100)
    
    # Generate a random displayed rating between 0 and 4111
    R = random.randint(0, 4111)
    
    return N, R

# Example of generating a test case
if __name__ == "__main__":
    N, R = tcgen()
    print(f"{N} {R}")
```

This generator will produce test cases within the specified constraints:
- \( N \) will be a random integer between 1 and 100.
- \( R \) will be a random integer between 0 and 4111.

You can run this function to generate random test cases for the problem. Here's how it works in practice:

1. **Generate a Random Number of Contests**: The number of contests \( N \) is randomly chosen between 1 and 100.
2. **Generate a Random Displayed Rating**: The displayed rating \( R \) is randomly chosen between 0 and 4111.

Output the generated test case in the required format. You can generate as many test cases as needed by calling the `tcgen()` function multiple times.