NL-NWO 2016 Problem 5

Bas has coloured each of the positive integers. He had several colours at his disposal. His colouring satises the following requirements: • each odd integer is coloured blue, • each integer $n$ has the same colour as $4n$ , • each integer $n$ has the same colour as at least one of the integers $n+2$ and $n + 4$ . Prove that Bas has coloured all integers blue.