MathDB

1#

Part of 2020 Middle European Mathematical Olympiad

Problems(1)

for what $k$ there are functions $f$,$g$

Source: 2020 MEMO I-1

8/30/2020
Let N\mathbb{N} be the set of positive integers. Determine all positive integers kk for which there exist functions f:N→Nf:\mathbb{N} \to \mathbb{N} and g:N→Ng: \mathbb{N}\to \mathbb{N} such that gg assumes infinitely many values and such that fg(n)(n)=f(n)+k f^{g(n)}(n)=f(n)+k holds for every positive integer nn.
(Remark. Here, fif^{i} denotes the function ff applied ii times i.e fi(j)=f(f(…f(j)… ))f^{i}(j)=f(f(\dots f(j)\dots )).)
algebranumber theorymemoMEMO 2020