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China mathematical olympiad 1995 problem2

Source: China mathematical olympiad 1995 problem2

September 15, 2013
functionalgebra unsolvedalgebrafunctional equation

Problem Statement

Let f:N→Nf: \mathbb{N} \rightarrow \mathbb{N} be a function satisfying the following conditions: (1) f(1)=1f(1)=1; (2) ∀n∈N\forall n\in \mathbb{N}, 3f(n)f(2n+1)=f(2n)(1+3f(n))3f(n) f(2n+1) =f(2n) ( 1+3f(n) ); (3) ∀n∈N\forall n\in \mathbb{N}, f(2n)<6f(n)f(2n) < 6 f(n). Find all solutions of equation f(k)+f(l)=293f(k) +f(l)=293, where k<lk<l. (N\mathbb{N} denotes the set of all natural numbers).
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