California Civil Code § 1350 et seq. appears to govern condominium associations such as yours, but it doesn't seem to answer this question.
Does the property management company have a written contract? If not, it could be argued that their position depends upon the continued assent of a majority of the owners. Another argument you could make is that the previous set of members could not create a contract longer than the duration of their several ownership interests. So, for example, if "Mr X", a previous owner, voted to hire the management company, his vote would be effective only as long as he owned one of the condos.
If you subscribe to this reasoning, the management company's contract lasts only as long as a majority of owners agree that it should last.
I was suggesting arguments you could use to terminate the management company despite the tie (which I gather you want to do).
But now that you have indicated the contract was for a fixed time, and that time has expired, you have a stronger case. Their contract is expired, and you would need a majority to take new action, either to approve a new contract or to approve payment without a contract.
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