5. There are 2001 coins on the table. Two players play the following game: they take turns, on each turn the first player can take any odd number of coins from 1 to 99, the second player can take any even number of coins from 2 to 100, and so on. The player who cannot make a move loses. Who wins with correct play?
5. The first one wins. Winning strategy: the first player should take 81 coins from the table on the first move. On each subsequent move, if the second player takes x coins, then the first player should take (101 - x) coins. He can always do this because if x is an even number from 2 to 100, then (101 - x) is an odd number from 1 to 99. Since 2001 = 101 * 19 + 81 + 1, after 19 such "responses" the first player will leave 1 coin on the table after his move, and the second player will not be able to make a move, i.e., will lose.