A. 565. The positive integers are coloured with a finite number of colours. A function f from the set of positive integers to itself has the following two properties:
(a) if xy, then f(x)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).
Does it follow that the function is bounded from above?
(Based on Romanian Master in Mathematics, problem 2012/3)
Deadline expired on 11 June 2012.