ELMO SL 2013 C8

There are 20 people at a party. Each person holds some number of coins. Every minute, each person who has at least 19 coins simultaneously gives one coin to every other person at the party. (So, it is possible that $A$ gives $B$ a coin and $B$ gives $A$ a coin at the same time.) Suppose that this process continues indefinitely. That is, for any positive integer $n$ , there exists a person who will give away coins during the $n$ th minute. What is the smallest number of coins that could be at the party?

Proposed by Ray Li