Learn Turkish: Yarışmada kaçıncı oldun? - What place did you get in the competition?

It is built from the question word kaç (how many) and the ordinal suffix -ıncı (which makes words like first, second, etc.). Literally, it translates to something like "the how-manieth?". Turkish uses this specific word whenever you ask for a number in a sequence, like rankings, grades, or floors in a building.

Why does the sentence use oldun instead of a verb like "get" or "finish"?

In English, you "get a place" or "come in first", but in Turkish, you use the verb olmak (to be / to become) for rankings. Oldun is the past tense for "you" (you became). To answer this question, you would say something like birinci oldum (I became first / I got first place).

Can I say hangi yer (which place) instead of kaçıncı?

It is much more natural and correct to use kaçıncı for rankings. Hangi (which) is used for general choices among items, whereas kaçıncı is specifically designed to ask for a numerical position in an ordered list.

How is Yarışmada formed?

The root word is yarışma, meaning "competition" or "contest". The suffix -da is the locative case, meaning "in" or "at". Together, it means "in the competition".