MathDB

Each positive integer is coloured red or blue. A function

f

from the set of positive integers to itself has the following two properties:
(a) if

x\le y

, then

f(x)\le f(y)

; and (b) if

x,y

and

z

are (not necessarily distinct) positive integers of the same colour and

x+y=z

, then

f(x)+f(y)=f(z)

.
Prove that there exists a positive number

a

such that

f(x)\le ax

for all positive integers

x

.
(United Kingdom) Ben Elliott

3