CGMO 2018 Problem 4

There're $n$ students whose names are different from each other. Everyone has $n-1$ envelopes initially with the others' name and address written on them respectively. Everyone also has at least one greeting card with her name signed on it. Everyday precisely a student encloses a greeting card (which can be the one received before) with an envelope (the name on the card and the name on envelope cannot be the same) and post it to the appointed student by a same day delivery. Prove that when no one can post the greeting cards in this way any more: (i) Everyone still has at least one card; (ii) If there exist $k$ students $p_1, p_2, \cdots, p_k$ so that $p_i$ never post a card to $p_{i+1}$ , where $i = 1,2, \cdots, k$ and $p_{k+1} = p_1$ , then these $k$ students have prepared the same number of greeting cards initially.