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all numbers coloured bw, if a,b white then a-b and a+b have diff. colours

Source: Dutch NMO 2011 p5

September 6, 2019

Coloringcombinatorics

Problem Statement

The number devil has coloured the integer numbers: every integer is coloured either black or white. The number

1

is coloured white. For every two white numbers

a

and

b

(

a

and

b

are allowed to be equal) the numbers

a-b

and

a +

b have different colours. Prove that

2011

is coloured white.