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f(1/2)=1975/ 2^{1975}

Source: Polish MO Finals 1975 p6

August 23, 2024
algebra

Problem Statement

On the interval [0,1][0,1] are given functions S(x)=1−xS(x) = 1 - x and T(x)=x/2T(x) = x/2. Does there exist a function of the form f=g1∘g2∘...∘gnf = g_1\circ g_2\circ ... \circ g_n, where n∈Nn \in N and each gkg_k is either S(x)S(x) or T(x)T(x), such that f(12)=197521975 ?f\left(\frac12\right)=\frac{1975}{2^{1975}} \, ?
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