Why do we repeat out loud the verses of Avinu Malkeinu - from Hachzireinu B'Teshuva until Kosveinu B'Sefer Slicha U'Mechila? Why only these verses? Why not others?
The custom of reciting Avinu Malkeinu repetitively by the chazzan and congregation is brought in Mateh Efrayim (תרב סי"ג), without any explanation. [The Mateh Efrayim himself writes to begin from א"מ קרע, however the Ktzey Hamateh at the bottom brings the custom to start from א"מ החזירנו בתשובה]. The only reason I could find is mentioned in שער יששכר מאמר מאזנים למשפט אות צא: During עשרת ימי תשובה we say א"מ כתבנו, whereas on a fast day we say א"מ זכרנו and during נעילה we say א"מ חתמינו. This could lead to confusion and therefore the chazzan says these verses loudly. (The שער יששכר himself writes that this is an unsatisfactory explanation, and in fact is of the opinion not the recite any verses repetitively).
However this does not explain why to begin from א"מ החזירנו בתשובה. The מועדים וזמנים ח"ו ס"ב explains that if we would begin from א"מ כתבנו לחיים it would appear as if all we are interested in is our own needs (as the Zohar says: "These dogs that say 'Give us life', 'Give us food'), and so we instead begin aloud by requesting that Hashem help us do Teshuva.