```python
def tint_mix(volume_a: int, strength_a: int, volume_b: int, strength_b: int) -> int:
    if not all(isinstance(i, int) for i in [volume_a, strength_a, volume_b]):
        raise ValueError("All inputs must be integers.")
    
    if not all(0 <= i <= 100 for i in [strength_a, strength_b]):
        raise ValueError("Strengths must be between 0 and 100 inclusive.")
    
    if volume_a == 0 and volume_b == 0:
        return 0
    
    total_volume = volume_a + volume_b
    total_strength = (volume_a * strength_a + volume_b * strength_b) // total_volume
    
    return total_strength
```