5

Part of 2020 May Olympiad

Problems(2)

$n$ is circular

Source: May Olimpiad 2020 L2 P5

We say that a positive integer

n

is circular if it is possible to place the numbers

1, 2, \cdots , n

in a circumference so that there are no three adjacent numbers whose sum is a multiple of 3. a) Show that 9 is not circular b) Show that any integer greater than 9 is circular.

Operation is to choose 4 cards and turn them over

Source: May Olimpiad 2020 L1 P5

On a table there are several cards, some face up and others face down. The allowed operation is to choose 4 cards and turn them over. The goal is to get all the cards in the same state (all face up or all face down). Determine if the objective can be achieved through a sequence of permitted operations if initially there are: a) 101 cards face up and 102 face down; b) 101 cards face up and 101 face down.