OMCC 2013 Problem 2

Around a round table the people $P_1, P_2,..., P_{2013}$ are seated in a clockwise order. Each person starts with a certain amount of coins (possibly none); there are a total of $10000$ coins. Starting with $P_1$ and proceeding in clockwise order, each person does the following on their turn:

• If they have an even number of coins, they give all of their coins to their neighbor to the left.
• If they have an odd number of coins, they give their neighbor to the left an odd number of coins (at least $1$ and at most all of their coins) and keep the rest.

Prove that, repeating this procedure, there will necessarily be a point where one person has all of the coins.