Sure, I can help write a test case generator for the given problem. Here's the Python code to generate test cases for the problem:

```python
import random

def generate_test_case():
    n = random.randint(2, 10**5)
    if n % 2 != 0:
        n += 1  # Ensure that n is even

    v = [random.randint(1, 10**5) for _ in range(n)]
    
    return n, v

# Function to format the test case for output
def format_test_case(n, v):
    return f"{n}\n{' '.join(map(str, v))}"

# Example of generating and printing a test case
if __name__ == "__main__":
    n, v = generate_test_case()
    print(format_test_case(n, v))

```

### Explanation:

1. **n Generation**: We generate `n` such that `2 ≤ n ≤ 10^5` and ensure `n` is even.
2. **v Generation**: We generate a list `v` of size `n` where each element is an integer between `1` and `10^5`.
3. **Formatting**: The function `format_test_case` takes the generated `n` and `v` and formats them for output.
4. **Example Usage**: The main block demonstrates how to generate and print a test case.

This generator will create random valid input test cases for the problem statement provided.