IMO Shortlist 2009 problem N3
Dodao/la:
arhiva2. travnja 2012. Let
![f](/media/m/9/9/8/99891073047c7d6941fc8c6a39a75cf2.png)
be a non-constant function from the set of positive integers into the set of positive integer, such that
![a-b](/media/m/3/0/c/30c4dc4baca89d41475d253434e82468.png)
divides
![f\!\left(a\right)-f\!\left(b\right)](/media/m/f/e/e/fee0337ba1f08875e60880b8a959c3df.png)
for all distinct positive integers
![a](/media/m/6/d/2/6d2832265560bb67cf117009608524f6.png)
,
![b](/media/m/e/e/c/eec0d7323095a1f2101fc1a74d069df6.png)
. Prove that there exist infinitely many primes
![p](/media/m/1/c/8/1c85c88d10b11745150467bf9935f7de.png)
such that
![p](/media/m/1/c/8/1c85c88d10b11745150467bf9935f7de.png)
divides
![f\!\left(c\right)](/media/m/6/8/b/68bdb159cffff89831ee526022fadb3b.png)
for some positive integer
![c](/media/m/e/a/3/ea344283b6fa26e4a02989dd1fb52a51.png)
.
Proposed by Juhan Aru, Estonia
%V0
Let $f$ be a non-constant function from the set of positive integers into the set of positive integer, such that $a-b$ divides $f\!\left(a\right)-f\!\left(b\right)$ for all distinct positive integers $a$, $b$. Prove that there exist infinitely many primes $p$ such that $p$ divides $f\!\left(c\right)$ for some positive integer $c$.
Proposed by Juhan Aru, Estonia
Izvor: Međunarodna matematička olimpijada, shortlist 2009