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Problem 6 — Cash Can

Source: 46th Austrian Mathematical Olympiad National Competition Part 2 Problem 6

July 14, 2018
Austrianumber theory

Problem Statement

Max has 20152015 jars labeled with the numbers 11 to 20152015 and an unlimited supply of coins.
Consider the following starting configurations:
(a) All jars are empty. (b) Jar 11 contains 11 coin, jar 22 contains 22 coins, and so on, up to jar 20152015 which contains 20152015 coins. (c) Jar 11 contains 20152015 coins, jar 22 contains 20142014 coins, and so on, up to jar 20152015 which contains 11 coin.
Now Max selects in each step a number nn from 11 to 20152015 and adds nn to each jar except to the jar nn.
Determine for each starting configuration in (a), (b), (c), if Max can use a finite, strictly positive number of steps to obtain an equal number of coins in each jar.
(Birgit Vera Schmidt)
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