MMM2019 Solutions.pdf
(1.35 MB, PDF)
… as follows. d = 0: 0, 0, 0, 0, 0 d = 1: 1, 0, 0, 0, 0 d = 2: 2, 0, 0, 0, 0 d = 3: 3, 0, 0, 0, 0 d = 4: 4, 0, 0, 0, 0 d = 5: 1, 1, 1, 1, 1 d = 6: 2, 1, 1, 1, 1 d = 7: 3, 1, 1, 1, 1 d = 8: 4, 1, 1, 1, 1 d = 9: 5, 1, 1, 1, 1 and in general for d = 5x+ y, with x = 0, 1, . . . and y = 0, 1, 2, 3, 4: d = 5x+ 0: x+ 0, x, x, x, x d = 5x+ 1: x+ 1, x, x, x, x d = 5x+ 2: x+ 2, x, x, x, x d = 5x+ 3: x+ 3, x, x, x, x d = 5x+ 4: x+ 4, x, x, x, x This way we use 21 balls of weight 0, 22 balls of weight 1, 23 balls … Han does not have it). Then, for each card he has played, Han knows the card Lando needs to have to win at the next turn. Moreover, Han knows the cards of Lando (all the numbers between 1 and 24 except the cards he had at the beginning and the cards they played). As Lando started Han has one more card than Lando when he plays, so he can choose a card such that Lando cannot win the next turn. Hence, Lando cannot win. Then, either Han wins before he plays his last card, or he plays his last card, and the sum 1 + 2 + ...+ 24 = 24 ⇤ 25/2 is divisible by 25. So Han …