Problem

Source: Romania National Olympiad 2015, grade x, p.3

Tags: function, algebra



Find all functions $ f,g:\mathbb{Q}\longrightarrow\mathbb{Q} $ that verify the relations $$ \left\{\begin{matrix} f(g(x)+g(y))=f(g(x))+y \\ g(f(x)+f(y))=g(f(x))+y\end{matrix}\right. , $$for all $ x,y\in\mathbb{Q} . $