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Interesting functional equation as P3 in the first-ever PAGMO

Source: Pan-American Girls’ Mathematical Olympiad, P3

October 5, 2021

Problem Statement

Let R\mathbb{R} be the set of real numbers. Determine all functions f:R⟶Rf: \mathbb{R}\longrightarrow \mathbb{R} so that the equality f(x+yf(x+y))+xf(x)=f(xf(x+y+1))+y2f(x+yf(x+y)) +xf(x)= f(xf(x+y+1))+y^2 is true for any real numbers x,yx,y.
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