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f(x^2+f(y))=f(f(y)-x^2)+f(xy)

Source: Own. IMO 2021 Malaysian Training Camp 1

December 31, 2020
algebra

Problem Statement

Find all continuous functions f:R→R f : \mathbb{R} \rightarrow \mathbb{R} such that for all real numbers x,y x, y f(x2+f(y))=f(f(y)−x2)+f(xy) f(x^2+f(y))=f(f(y)-x^2)+f(xy)
[Extra: Can you solve this without continuity?]
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