7

Part of 2024 Assara - South Russian Girl's MO

Problems(2)

Chip on a board makes moves

Source: Assara 2024 J7

There is a chip in one of the squares on the checkered board. In one move, she can move either

1

square to the right, or diagonally

1

to the left and

1

1

to the left and

3

down (see Fig.). The chip made

n

moves and returned to the starting square. Prove that a)

n

is divisible by

2

n

is divisible by

8

n|(a+b)^2,(b+c)^2,(c+a)^2

Source: Assara 2024 S7

Find all positive integers

n

for such the following condition holds: "If

a

b

c

are positive integers such are all numbers

a^2+2ab+b^2,\ b^2+2bc+c^2, \ c^2+2ca+a^2

are divisible by

n

(a+b+c)^2

is also divisible by

n

." G.M.Sharafetdinova