Question 7
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 $f(x) \le a x$ for all positive integers $x$.
(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 $f(x) \le a x$ for all positive integers $x$.