Mathematical and Physical Journal
for High Schools
Issued by the MATFUND Foundation
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Problem A. 565. (May 2012)

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 x\ley, then f(x)\lef(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 \frac{f(x)}x is bounded from above?

(Based on Romanian Master in Mathematics, problem 2012/3)

(5 pont)

Deadline expired on June 11, 2012.


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Problems in Mathematics of KöMaL, May 2012