The state music contest, a competition that happens once a year, had arrived.
Explanation:
I have been able to rewrite sentences 3 and 4 as one sentence containing an appositive phrase.
An appositive phrase refers to a noun or a noun phrase that actually renames the noun next to it. The appositive phrase goes ahead to explain and give more meaning to the noun next to it.
In the above sentence, "<em>a competition that happens once a year</em>" is the appositive phrase that explains or renames "<em>The state music contest</em>".
The purpose it actually serves is to give more information about the noun.
