Algebra · Functional equations · Reciprocal substitution · Mobius maps · Range of a function
Determine all functions \(f, g \colon \left(\tfrac{1}{2}, 2\right) \to \mathbb{R}\) such that for every \(x \in \left(\tfrac{1}{2}, 2\right)\),
\[ x f(x) + g\!\left(\frac{4x+1}{2x+2}\right) = x \qquad \text{and} \qquad 2 f\!\left(\frac{1}{x}\right) - g\!\left(\frac{x+4}{2x+2}\right) = -4x. \]
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Serbian Regional Competition (Okruzno takmicenje) 2022, high school grade I, category A, problem 2. Organized by the Mathematical Society of Serbia (DMS). Source