2014 IMO Shortlist C8

A card deck consists of 1024 cards. On each card, a set of distinct decimal digits is written in such a way that no two of these sets coincide (thus, one of the cards is empty). Two players alternately take cards from the deck, one card per turn. After the deck is empty, each player checks if he can throw out one of his cards so that each of the ten digits occurs on an even number of his remaining cards. If one player can do this but the other one cannot, the one who can is the winner; otherwise a draw is declared. Determine all possible first moves of the first player after which he has a winning strategy. (Russia)

一副牌由 1024 张牌组成。在每张卡上，写入一组不同的十进制数字，其中没有两组数字重合(因此，其中一张卡是空的)。两名玩家轮流从牌堆中拿牌，每回合一张牌。当这副牌空了之后，每个玩家检查他是否可以扔掉一张牌，以便十个数字中的每一个出现在他剩余的牌上的偶数张上。如果一名玩家能做到这一点而另一名玩家不能，则能做到的一方获胜；否则宣布平局。确定第一个玩家的所有可能的第一步，然后他就有了获胜的策略。 (俄罗斯)