BALKAN 2024 Problem 4

Let $\mathbb{R}^{+}=(0, \infty)$ be the set of all positive real numbers. Find all functions $f: \mathbb{R}^{+} \rightarrow \mathbb{R}^{+}$ and polynomials $P(x)$ with non-negative real coefficients such that $P(0)=0$ which satisfy the equality

$$f(f(x)+P(y))=f(x-y)+2 y$$

for all real numbers $x>y>0$.