How do i prove i didnt do property damages

Without more information it does appear that water drains downward and if the area of damage in the ceiling is under your apartment you need to inspect and discuss with your landlord any evidence of leakage.  Unless you caused the water to over flow or otherwise negilgently allowed the leak, the landlord is responsible for maintaining plumbing fixtures.    You may need to hire a plumber to inspect if the landlord refuses to do so.  I would write the landlord that you may seek to take any costs of inspection and repair off your rent if the plumbing is faulty.   Under Hawaii landlord tenant law the landlord is responsible for plumbing unless the issue is the result of abuse or tenant negligence such as leaving water running or not requesting repairs in a timely fashion.   Of course your written landlord tenant ageement should be inspected for terms that may cover this situation.