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R+ FE with arbitrary constant

Source: APMO 2023/4

July 5, 2023
APMOAPMO 2023functional equationfe

Problem Statement

Let c>0c>0 be a given positive real and R>0\mathbb{R}_{>0} be the set of all positive reals. Find all functions f ⁣:R>0→R>0f \colon \mathbb{R}_{>0} \to \mathbb{R}_{>0} such that f((c+1)x+f(y))=f(x+2y)+2cx   \textrm{for all } x,y \in \mathbb{R}_{>0}.
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