IMO Shortlist 2010 problem A6
Dodao/la:
arhiva23. lipnja 2013. Suppose that
![f](/media/m/9/9/8/99891073047c7d6941fc8c6a39a75cf2.png)
and
![g](/media/m/9/5/8/958b2ae8c90cadb8c953ce50efb9c02a.png)
are two functions defined on the set of positive integers and taking positive integer values. Suppose also that the equations
![f(g(n)) = f(n) + 1](/media/m/1/2/f/12fdb329b435e83d2c8b094de0bd8aa8.png)
and
![g(f(n)) = g(n) + 1](/media/m/8/5/9/8593634a0dae1a575f250b04ca77d725.png)
hold for all positive integers. Prove that
![f(n) = g(n)](/media/m/3/0/a/30a9050e30d28b41269edac6d2823b93.png)
for all positive integer
![n.](/media/m/4/7/8/478d25bfe04800537ae6e85be9d11ea2.png)
Proposed by Alex Schreiber, Germany
%V0
Suppose that $f$ and $g$ are two functions defined on the set of positive integers and taking positive integer values. Suppose also that the equations $f(g(n)) = f(n) + 1$ and $g(f(n)) = g(n) + 1$ hold for all positive integers. Prove that $f(n) = g(n)$ for all positive integer $n.$
Proposed by Alex Schreiber, Germany
Izvor: Međunarodna matematička olimpijada, shortlist 2010