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1 <= f(x)-x<= 2019, f(f(x))= x mod 2019 , f^k(x)=x+2019 k

Source: 2019 Nigerian Senior Mathematics Olympiad Round 4 (final) problem 1

September 9, 2019
functionfunctional equationnumber theoryinequalities

Problem Statement

Let f:N→Nf: N \to N be a function satisfying (a) 1≤f(x)−x≤20191\le f(x)-x \le 2019 ∀x∈N\forall x \in N (b) f(f(x))≡xf(f(x))\equiv x (mod 20192019) ∀x∈N\forall x \in N Show that ∃x∈N\exists x \in N such that fk(x)=x+2019k,∀k∈Nf^k(x)=x+2019 k, \forall k \in N
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