IMO Shortlist 1995 problem S6
Dodao/la:
arhiva2. travnja 2012. Let
![\mathbb{N}](/media/m/6/0/7/607ae4ca194aaec986c3e218df0f3079.png)
denote the set of all positive integers. Prove that there exists a unique function
![f: \mathbb{N} \mapsto \mathbb{N}](/media/m/0/6/4/064f5e79a7f8bd81b5e7163e70e87d6b.png)
satisfying
for all
![m](/media/m/1/3/6/1361d4850444c055a8a322281f279b39.png)
and
![n](/media/m/a/e/5/ae594d7d1e46f4b979494cf8a815232b.png)
in
![\mathbb{N}.](/media/m/c/1/b/c1b4c32cab52be62370184297322cb70.png)
What is the value of
%V0
Let $\mathbb{N}$ denote the set of all positive integers. Prove that there exists a unique function $f: \mathbb{N} \mapsto \mathbb{N}$ satisfying
$$f(m + f(n)) = n + f(m + 95)$$
for all $m$ and $n$ in $\mathbb{N}.$ What is the value of $\sum^{19}_{k = 1} f(k)?$
Izvor: Međunarodna matematička olimpijada, shortlist 1995