Problem 4, 2014

Three piles of tokens lie on a table, holding \(a\), \(b\) and \(c\) tokens, where \(a \ge b \ge c > 0\). Players \(A\) and \(B\) move tokens alternately, and \(A\) starts.

In one move a player first selects two of the piles, and then transfers at least one token from the pile holding fewer tokens to the pile holding more. If the two selected piles hold equally many tokens, the player transfers at least one token from either of them to the other.

The player after whose move all the tokens lie on a single pile wins. Determine, in terms of \(a\), \(b\) and \(c\), which player has a winning strategy.