KöMaL 2020 B5140

There are 10 countries on an island, some of which share a border, and some do not. Each country uses a currency of its own. Every country operates a single exchange office, by the following rules: if you pay 10 units of the currency of that country, you will get 1 unit of each of the currencies of the bordering countries. Initially Arthur and Theodore each own 100 units of the currency of every country. Then each of them shops around the exchange offices of various countries in any order they like, and keeps exchanging money while they can (that is, while they have at least 10 units of a kind). Prove that Arthur and Theodore will have the same number of Bergengocian dollars at the end (the Bergengocian dollar is the currency of one of the countries on the island).