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Unique function

Source: Serbia MO 2023 P5

April 29, 2023
functioncalculusalgebrafunctional equation

Problem Statement

Let f:R→Rf: \mathbb{R} \rightarrow \mathbb{R} be a function which satisfies the following: [*] f(m)=mf(m)=m, for all m∈Zm\in\mathbb{Z};[*] f(a+bc+d)=f(ac)+f(bd)2f(\frac{a+b}{c+d})=\frac{f(\frac{a}{c})+f(\frac{b}{d})}{2}, for all a,b,c,d∈Za, b, c, d\in\mathbb{Z} such that ∣ad−bc∣=1|ad-bc|=1, c>0c>0 and d>0d>0;[*] ff is monotonically increasing. (a) Prove that the function ff is unique. (b) Find f(5−12)f(\frac{\sqrt{5}-1}{2}).
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