5

Part of 2023 May Olympiad

Problems(2)

50 stacks of coins that have 1,2,3,…,50 coins respectively

Source: 2023 May Olympiad L2 p5

On the table there are

50

stacks of coins that have

1,2,3,…,50

coins respectively. Ana and Beto play the following game in turns: First, Ana chooses one of the

50

piles on the table, and Beto decides if that pile is for Ana or for him. Then, Beto chooses one of the

49

remaining piles on the table, and Ana decides if that pile is for her or for Beto. They continue playing alternately in this way until one of the players has

25

batteries. When that happens, the other player takes all the remaining stacks on the table and whoever has the most coins wins. Determine which of the two players has a winning strategy.

combinatoricsgamegame strategy

1000 numbered cards placed in 100 numbered boxes, removing digits

Source: 2023 May Olympiad L1 p5

100

boxes that were labeled with the numbers

00

01

02

…

99

000

001

002

…

999

were written on a thousand cards, one on each card. Placing a card in a box is permitted if the box number can be obtained by removing one of the digits from the card number. For example, it is allowed to place card

037

07

, but it is not allowed to place the card

156

65

.Can it happen that after placing all the cards in the boxes, there will be exactly

50

empty boxes? If the answer is yes, indicate how the cards are placed in the boxes; If the answer is no, explain why it is impossible

combinatoricsnumber theory