Learn Turkish: Şartname, adaylık için dil sınavını geçmiş olmayı istiyor. - The specification requires having passed the language exam for candidacy.

geçmiş olmayı is the perfective infinitive, meaning "to have passed." If we used geçmeyi ("to pass"), it could imply taking and passing the exam at the time of candidacy. geçmiş olmayı emphasizes that the passing must already be a completed action before applying.

Why does olma take the accusative ending -yı here?

It acts as the direct object of the main verb, istiyor (wants/requires). The specification wants what? It wants "having passed the exam" (geçmiş olmayı). Since istemek requires a specific direct object to be in the accusative case, we add the buffer -y- and the accusative -ı to the verbal noun olma.

Why is sınav written as sınavını?

First, dil sınavı is a noun compound meaning "language exam," giving us the third-person possessive suffix -ı. Then, because the exam is the specific direct object of the verbal phrase geçmiş olmayı, it needs the accusative case. We add the buffer -n- and the accusative -ı, resulting in sınavını.

Why don't we use a personal possessive ending on olma, like olmasını?