CF1147C Thanos Nim

Alice and Bob are playing a game with $ n $ piles of stones. It is guaranteed that $ n $ is an even number. The $ i $ -th pile has $ a_i $ stones. Alice and Bob will play a game alternating turns with Alice going first. On a player's turn, they must choose exactly $ \frac{n}{2} $ nonempty piles and independently remove a positive number of stones from each of the chosen piles. They can remove a different number of stones from the piles in a single turn. The first player unable to make a move loses (when there are less than $ \frac{n}{2} $ nonempty piles). Given the starting configuration, determine who will win the game.

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In the first example, each player can only remove stones from one pile ( $ \frac{2}{2}=1 $ ). Alice loses, since Bob can copy whatever Alice does on the other pile, so Alice will run out of moves first. In the second example, Alice can remove $ 2 $ stones from the first pile and $ 3 $ stones from the third pile on her first move to guarantee a win.