Learn Polish: Panie doktorze, mam problem.

In Polish, it is standard and polite to include Panie (sir) or Pani (madam) before a professional title when talking directly to someone. Dropping it and just saying Doktorze sounds very direct and can be seen as less polite in a formal setting.

How does the word doktor change to doktorze in this sentence?

Since you are addressing the doctor directly, you must use the Vocative case. Masculine nouns ending in a hard consonant like 'r' usually take an '-e' ending. When we add this '-e', the 'r' changes to the softer 'rz' sound to make it easier to pronounce, giving us doktorze.

Why is there an 'i' in Panie when the basic word is pan?

Just like the doctor's title, the word pan takes an '-e' ending in the Vocative case. When adding '-e' to an 'n', the consonant softens. In Polish spelling, we write this softened sound as 'nie', which is why pan becomes Panie.

The verb mam requires the Accusative case. Why doesn't the word problem change its ending?

You are right that mieć (to have) requires the Accusative case for its object. However, problem is a masculine inanimate noun (a thing, not a living being). For these types of nouns, the Accusative form is exactly the same as their basic dictionary form.