1

Part of 2019 Bundeswettbewerb Mathematik

Problems(2)

Chessboard covering with dominoes. Is there a square?

Source: Bundeswettbewerb Mathematik 2019, Round 1 - Problem 1

8 \times 8

chessboard is covered completely and without overlaps by

32

dominoes of size

1 \times 2

. Show that there are two dominoes forming a

2 \times 2

combinatoricscombinatorial geometry

120 pirates split 119 gold pieces, no more than 14 each

Source: Germany Federal - Bundeswettbewerb Mathematik 2019 round 2, p1

120

pirates distribute

119

gold pieces among themselves. Then the captain checks if any pirate has

15

or more gold pieces. If he finds the first one, he must give all his gold pieces to other pirates, whereby he may not give more than one gold piece to anyone. This control is repeated as long as there is any pirate with

15

or more gold pieces. Does this process end after a lot of checks?