Certainly! Here is the test case generator for the problem statement you provided:

```python
import random

def generate_large_integer(max_digits):
    # Generate a random number with a specific number of digits
    return ''.join(random.choices('123456789', k=1) + random.choices('0123456789', k=max_digits-1))

def tcgen():
    # Maximum number of digits is 100
    max_digits = 100
    
    # Generate random lengths for A and B
    len_A = random.randint(1, max_digits)
    len_B = random.randint(1, max_digits)
    
    # Generate the numbers with the specified lengths
    A = generate_large_integer(len_A)
    B = generate_large_integer(len_B)
    
    return A, B

# Example usage:
A, B = tcgen()
print(A)
print(B)
```

### Explanation:
1. **Generating Large Integers**:
    - `generate_large_integer`: This function generates a large integer represented as a string with a specified number of digits. The first digit is chosen from '1' to '9' to ensure it does not begin with '0'.
    - The rest of the digits are chosen from '0' to '9'.

2. **Test Case Generator**:
    - `tcgen`: This function randomly determines the lengths of integers \(A\) and \(B\) (up to 100 digits each).
    - Uses `generate_large_integer` to create \(A\) and \(B\) with the specified lengths.
    - Returns \(A\) and \(B\) as strings.

### Running the Generator
- When you run the generator, it prints the two large integers \(A\) and \(B\). You can use these values as input for your competitive programming problem.

### Sample Output:
Below is an example of what the output might look like when you run the generator:

```
2754893274018947583947501984750198475019847501984750198475019847501984750198475019847501984750198475
8750198475019847501984750198475019847501984750198475019847501984750198475019847501984750198475019847
```