Question 14
There are three piles of coins, with $a$, $b$ and $c$ coins respectively, where $a$, $b$, $c \ge 2015$ are positive integers. The following operations are allowed:
(1) Choose a pile with an even number of coins and remove all coins from this pile. Add coins to each of the remaining two piles with amount equal to half of that removed; or
(2) Choose a pile with an odd number of coins and at least $2017$ coins. Remove $2017$ coins from this pile. Add $1009$ coins to each of the remaining two piles.
Suppose there are sufficiently many spare coins. Find all ordered triples $(a, b, c)$ such that after some finite sequence of allowed operations, there exists a pile with at least $2017^{2017}$ coins.
(1) Choose a pile with an even number of coins and remove all coins from this pile. Add coins to each of the remaining two piles with amount equal to half of that removed; or
(2) Choose a pile with an odd number of coins and at least $2017$ coins. Remove $2017$ coins from this pile. Add $1009$ coins to each of the remaining two piles.
Suppose there are sufficiently many spare coins. Find all ordered triples $(a, b, c)$ such that after some finite sequence of allowed operations, there exists a pile with at least $2017^{2017}$ coins.