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real functions which map the sum of n elements into the sum of their image

Source: 1984 Polish MO Finals p1

February 25, 2020
functionalgebraSumcomposition

Problem Statement

Find the number of all real functions ff which map the sum of nn elements into the sum of their images, such that fn−1f^{n-1} is a constant function and fn−2f^{n-2} is not. Here f0(x)=xf^0(x) = x and fk=f∘fk−1f^k = f \circ f^{k-1} for k≥1k \ge 1.
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