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Functional equation with polynomial from R^+ to R^+

Source: BMO 2024 Problem 4

April 29, 2024
algebrafunctional equationpolynomial

Problem Statement

Let R+=(0,∞)\mathbb{R}^+ = (0, \infty) be the set of all positive real numbers. Find all functions f:R+→R+f : \mathbb{R}^+ \to \mathbb{R}^+ and polynomials P(x)P(x) with non-negative real coefficients such that P(0)=0P(0) = 0 which satisfy the equality f(f(x)+P(y))=f(x−y)+2yf(f(x) + P(y)) = f(x - y) + 2y for all real numbers x>y>0x > y > 0.
Proposed by Sardor Gafforov, Uzbekistan
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