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Found problems: 2

2020 Hong Kong TST, 4
Tags: algebra , TST , 2020 , hong Kong TST , fuctional equation , fe , Reals
Find all real-valued functions $f$ defined on the set of real numbers such that $$f(f(x)+y)+f(x+f(y))=2f(xf(y))$$ for any real numbers $x$ and $y$.

2021 Iran MO (3rd Round), 3
Tags: algebra , polynomial , function , fuctional equation
Polynomial $P$ with non-negative real coefficients and function $f:\mathbb{R}^+\to \mathbb{R}^+$ are given such that for all $x, y\in \mathbb{R}^+$ we have $$f(x+P(x)f(y)) = (y+1)f(x)$$ (a) Prove that $P$ has degree at most 1. (b) Find all function $f$ and non-constant polynomials $P$ satisfying the equality.