2

Part of 2021 May Olympiad

Problems(2)

special and common numbers

Source: May Olympiad 2021 L2 #2

N

be a positive integer; a divisor of

N

is called common if it's great than

1

and different of

N

. A positive integer is called special if it has, at least, two common divisors and it is multiple of all possible differences between any two of their common divisors. Find all special integers.

2 colours in a 2x8 board (2021 May Olympiad L1 p2)

Source:

2 \times 8

squared board, you want to color each square red or blue in such a way that on each

2 \times 2

sub-board there are at least

3

boxes painted blue. In how many ways can this coloring be done?
Note. A

2 \times 2

board is a square made up of

4

squares that have a common vertex.

combinatoricsColoring