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Functional Equation Pair Over Reals

2425MasterFunctionsAlgebra

二〇一二數學奧林匹亞競賽第一階段選訓營

Let $f$ and $g$ be functions from the real numbers to the real numbers. Find all ordered pairs $(f, g)$ of functions satisfying the condition $$ g(f(x + y)) = f(x) + (2x + y)g(y), \text{ for all real numbers } x, y. $$
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