IMO Shortlist 2014 problem C8

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Dodao/la: arhiva
7. svibnja 2017.

A card deck consists of 1024 cards. On each card, a set of distinct decimal digits is written is such a way that no two of these sets concide (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.